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Full text of "Shrinkage stresses in concrete."

Research Laboratory 

of the 

Portland Cement Association 

Bulletin 11 

Shrinkage Stresses in Concrete 

Part 1— Shrinkage (or Swelling), Its Effect upon Displacements 
and Stresses in Slabs and Beams of Homogeneous, Iso- 
tropic, Elastic Material 

Part 2 — Application of the Theory Presented in Part 1 to Experi- 
mental Results 

By 
GERALD PICKETT 



March, 1946 
( *m< AGO 



Authorized Reprint from Copyrighted 

Journal op the American' Concrete Institute 

New Center Building, 7400 Second Boulevard 

Detroit 2, Michigan 

Jm. and Feb. 1946, Proceedings Vol. 42, pp. 165-204 and 361-400 



Title 48-8 — o port of PROCEEDINGS, AMERICAN CONCRETE INSTITUTE Vol. 42 

JOURNAL 
AMERICAN CONCRETE INSTITUTE 

(copyrighted) 
Vol. 17 No. 3 7400 SECOND BOULEVARD, DETROIT 2, MICHIGAN January 1946 



Shrinkage Stresses in Concrete* 

By GERALD PICKETTt 

Member American Concrete Institute 

SYNOPSIS 

Theoretical ex] :ttions of < and slabs 

that occur during the course of drying and expressions for distribution 
of the accompanying shrinkage stresses are derived in Part 1. These 
expressions an on the assumption that the laws governing 

the development of shrinkage stresses in concrete during drying are 
analogous to those governing .-.pment of thermal stresses in 

an ideal body during cooling. Three cases are considered: 

(a) slab or beam drying from one face <>! 

(b) slab or beam drying from two oppo and 

(c) prism drying from four fat 

The applicability of the equations to concrete is considered in Part 
2 (to appear ACI Journ It is shown that the 

course of shortening of prisms is in very good agreement with the 
theoretical equations and that from a test on one prism the shortening 
versus period of diving of other prisms of the same mat. rial differing in 
size and number of sides exposed to drying can be predicted with fair 
accuracy if the differences in size are not too great. However, it is shown 
thai the theory must be modified to take into account inelastic deforma- 
tion and to permit the supposed constants of the mat .rial to vary with 
moisture content and size of sp if the theory is to be in agree- 

ment with all results on all types of specimen of a given concrete. 

Various tests are described which, when used in conjunction with 
the theory, provide a means foi g some of the more fundamental 

properties of concrete and for predicting the performance of concrete 
under some conditions in the field 

INTRODUCTION 

Concrete, like many other materials, gains or lost- nth changes 

in ambient conditions. With each change in water con 1 concrete 



♦Received by the Institute. April 30, 1945. 

1 Professor ot Applied Mechanic* Kansas State < oUege, Manhattan, kans.; formerly Portland Cement 
Association Research Laboratory, Chicago. 

(165) 



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SHRINKAGE STRESSES IN CONCRETE 167 

the distribution of shrinkage stresses in, concrete beams and slabs during 
the course of drying. 

Second, to show by means of data from specimens under controlled 
conditions the manner and degree to which the equations apply to con- 
crete. 

Third, to suggest methods for studying some of the more fundamental 
properties of drying concrete. 

No attempt will be made here to give a complete analysis of stresses in 
concrete. In particular, the effect of aggregate particles on the stresses 
within the hardened paste will not be considered. 

Before expressions for shrinkage stresses in concrete can be derived, 
assumptions must be made in regard to the relation between shrinkage 
and moisture content and the laws controlling the flow of moisture in 
concrete as well as the relation between stress and strain. 

The actual relationships are not as simple as could be desired. If the 
flow of water were entirely by vapor diffusion, if the vapor pressure of the 
water in the concrete were proportional to the moisture-content, and if 
permeability were independent of the moisture-content, then the differ- 
ential equation for the flow of water would be a partial-differential 
equation known in physics and mathematics either as the diffusion 
equation or as the equation of heat conduction. Carlson, 1 * in a study of 
distribution of moisture in concrete, assumed that this equation applies. 
If the flow of water could be expressed by the diffusion equation and if the 
shrinkage (or swelling) tendency! of each elemental volume were 
linearly related to the moisture-content, the unrestrained shrinkage (or 
swelling) could also be expressed by the diffusion equation. This possi- 
bility was also considered by Carlson. 1 But the flow of water is different 
from that indicated by the diffusion equation, and the relationship 
between the change in moisture-content and unrestrained shrinkage is not 
linear as required by these equations. Moreover, satisfactory expressions 
for either the flow of water or the moisture-shrinkage relation have not 
been found. 

It is believed that moisture in concrete flows partly as liquid in capillar- 
ies, partly as vapor, and partly as adsorbed liquid on the surface of the 
colloidal products of hydration. While drying progresses, the vapor 
pressure of the water remaining in the region losing water decreases 
progressively with the moisture content. This change in vapor pressure 
with change in moisture content is not linear with respect to moisture 



*See references at end of text of Part 1. 

fBy shrinkage (or swelling) tendency is meant the unit linear deformation due to any cause other than 
stress that would occur in an infinitesimal element if the element were unrestrained by neighboring elements. 
It is not to be confused with the average unit deformation, commonly called shrinkage, of a so-called un- 
restrained specimen, nor with the resultant linear unit deformation which for the x-direction will be de- 
signated ?z. Hereinafter, the linear unit shrinkage tendency will be referred to either as unrestrained 
shrinkage, for clarity, or merely as shrinkage, for brevity. 






JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



. ther i- the rate <«f How proportional to the gradient id \ 

and relative proportions of the sp upied by 

liquid : a drying proceeds. Thia fact, ae weX\ na the 

non-uniformit a believed to be partly responsible for the 

bich vapor pressure depends on moisture content and the way in 

win- ■■ gradient of vapor pressure. 

ght-loea relation is different for different 
ritioo of the concrete and the conditions 
, Bferenl during first shrinkage from 
subsequent volume changes, [fasatui 
the ratio <>i change of length to lo 
At first, comparatively Bmall cha 
jiit . The higher the water-cement 
I be smaller the change during 

' < ■ omes imi<-li larger and 
i i w bich ii may cither increase 

j,i [t is believed that 

■ bat bold 1 1n* \\ai. I e Bhape, 

cles; and (c) th< 'It oi the 

• red in any at udy 
concrete !«■' 
• «i.M'> i rom t In 

hrough t he si • 

^e> ImiI h 

;.prc»\ima1i T\ 
l:i\\ tO In- ;i — Mined, hilt 
m-, ill h< 

and foi 

1 \1< 

.nd \\hil< 






• 



SHRINKAGE STRESSES IN CONCRETE 169 

the diffusion equation applies to shrinkage even though the simple rela- 
tions that are implied by that assumption are contrary to fact. It is fur- 
ther assumed that concrete follows Hooke's law. The derivations given 
in Part 1 are based upon these assumptions. 

Since in Part 1 the derivations for deformations and stresses are based 
on the assumptions that shrinkage follows the diffusion equation and the 
material follows Hooke's law, the equations are even more applicable to 
thermal stresses in metals than to shrinkage stresses in concrete. In fact, 
much of the mathematical work given here was taken from the literature 
on diffusion of heat and on thermal stresses, as the references will show. 
However, certain corresponding coefficients in the two problems are of an 
entirely different order of magnitude. For example, the numerical value 
of the thermal diffusivity for steel expressed in square inches per second 
is approximately the same as the numerical value of the shrinkage diffu- 
sivity of concrete expressed in square inches per day. Because of the 
relatively slow diffusion of shrinkage the application of the hypothesis to 
the shrinkage of concrete necessitates the study of early transient condi- 
tions (usually ignored in the treatment of heat). 

PART 1— SHRINKING (OR SWELLING), ITS EFFECT UPON DISPLACEMENTS AND 
STRESSES IN SLABS AND BEAMS OF HOMOGENEOUS, ISOTROPIC, ELASTIC 

MATERIAL 

Notation 

S = free, unrestrained unit linear shrinkage-strain 

~S = free, unrestrained unit linear swelling-strain 

5 ro = final shrinkage-strain under fixed ambient conditions, value of S when t = a> 

S av = average shrinkage over the volume of the specimen, the same as average shorten- 
ing per unit length if the material follows Hooke's law 

t = time in days 

k = diffusivity coefficient of shrinkage in sq. in. per day 

/ = surface factor, characteristic of the material and the boundary conditions, in in. per 
day 

a, 6, c, d, I = distances related to the dimensions of the specimen in inches 

B = fb/k, a non-dimensional parameter 

T = kt/b 2 , a non-dimensional parameter 

B c and T c , non-dimensional parameters corresponding to B and T and used when a 
second characteristic dimension of the specimen must be considered 

x, y, z = rectangular coordinates 

n = nth root of ptan = B 

$ m = same as /3 n except used in connection with c, whereas n is used in connection 
with b 

A „ = Fourier coefficient 

2B „ 2£, 

F n = , F m =~ 



& + B + j3» Bl + B c + fil 

b -f„, a. -A 

$1 el 



H n = — F m ff m = — F„ 



170 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



V»A 2 J 



<r t = normal components of stress parallel tox-, y-, and z -axes— positive if 
tensile, negative if compressive. 

= elongations inx— , y— , and 2 -directions 
Txv, *». T r , = shearing-stress components 
y*,, 7*z, Ty» = shearing-strain components 
E = Young's modulus in 
u = Poisson's ratio 

r = deflection in inches, displacement of the elastic line in the y-direetion 
-N = the normal to the surface directed outward 



rX -x= 



*(*) = 



2 rl 

P(x) = — = j e fa t the probability integral 

2 r 00 _ X 2 

- -m 



fa = 1 — - 
1 



cos n - 





cc -7^ COS 0* - 

*< = 1 - - F m - 

1 cos &» 



H b 



0= -r« 

1 - - #. 

1 



H. = 1- 






Equation for diffusion of unrestrained shrinkage 

The diffusion equation is a mathematical statement of the fact that for 

each infinitesimal volume - the excess of the substance in question 

flowing in over that flowing out per unit of time is equal to the rate of 

mcr ; ft <* in that volume. When similar assumptions are 

k in regard to shrinkage, shrinkage thus being treated as if it were a 

;ust as heat is so treated, the result is- 



'' La* 1 ty 1 az' J " 

where fc is the diffusivity of shrinkage. 
The equation becomes 



..(1) 



k 5 -**—« 



2 



SHRINKAGE STRESSES IN CONCRETE 171 

at exposed boundaries and 

^= (3) 

dX 

at sealed boundaries 
wherr 

N is the normal to the surface, directed away from the body 

/ is the surface factor 

S oo is the value that S will eventually reach under fixed ambient 
conditions. 

Equations 2 and 3 correspond, respectively, to Newton's law of cooling 
at exposed boundaries and to no flow of heal 3 erfectly insulated 
boundaries in the analogous problem of flow of heat. 

If the boundaries of the body are not parallel planes, a transformation 
of Equation 1 from an expression in rectangular coordinates to some 
other form is usually desirable. For i . if the bod si circular 

cylinder, Equation 1 is best transformed to 

Ldr* r dr r 2 d9 2 d z- J dt 

where r, 6, and z are cylindrical coordinates. Frequently, the condition 

— = at some boundaries or some other conditions will make S inde- 

dX 

pendent of certain coordinates and thereby simplify Equation 1. 

Since the form of the solution for S depends upon the form of the 
differential equation, the form of the solution is dependent upon the 
boundary condition and the shape of the body under investigation. The 
initial conditions (values of S at t = 0) and any variation in boundary 
conditions with time will also affect the form of the solution. 

Assumptions as to elastic properties 

After a satisfactory solution for S has been obtained, then displace- 
ments and stresses will be found by the application of certain funda- 
mentals of the theory of elasticity. The solutions for stresses are here 
restricted to homogeneous isotropic solids that follow Booke'a law. 
Also, as will be brought out below, the effect of Poisson's ratio will be 
neglected in some cas 

Effect of shape of body on relative values of principal stresses 

The state of stress at any point in a body is defined by the directions 
and magnitudes of the three principal stresses. The three principal 
stresses in wide -labs and in narrow brain- will be in the directions of 
length, width, and depth, respectively, if the bodies are under uniform 
exposure either from one or from two opposite face- and are without 
external restraint. The principal stress in the direction of depth (normal 



• 



1946 



tresses will be 
D - MBMfled t<» «i«*pt It the prinri- 






I 



-« VM.ltl 






« - W ill Im 






M ON 















z 












SHRINKAGE STRFSSES IN CONCRETE 173 

Equation 2a is also satisfied if fi n is I he nth root of 

(3 tan 13= f - (2b) 

n 

i.e., j8 B tan Pn = ~ (2c) 

A 

The above statements may be verified by substituting S from Equation 
4 into Equations la, 2a and 3a. 

For time t = » , Equation 4 reduces to S = S co , which is in accord with 
the definition of 5 oo . An infinite series of terms such as the trigonometric 
series in Equation 4 is necessary to give an arbitrary distribution of 
shrinkage at time t = 0. 

If the initial conditions are such that S = when t = 0, then the 
Fourier coefficients A n are given by* 



^n = 



2 / " 6 

O co tC 



S n (fbV + Jg , 
U/ k 



cos$ n (fb\ 
It therefore follows that 



V 

COS^ 



s sr -t£ „ h 



Si 

where 



= 1- Te'S.-/ (5) 

£— cos p n 



F = 2B 

' n 



5 2 + B + ft 

Equation 5 (in slightly different form) and similar equations for other 
shapes and other conditions, applied to analogous phenomena, may be 
found in the literature of mathematical physics such as the textbooks 
of Byerly, Carslaw, and Ingersoll and Zobel. Various tables and diagrams 
have been prepared from which the numerical relationship of the four 
non-dimensional quantities S/S «, y/b, B, and T may be found, such as 
Fig. 4, page 841 of Perry's Chemical Engineer's Handbook (1934), 

* T he general procedure of obtaining Fourier coefficients to satisfy initial conditions semewhat analogous 
tO tin- present problem ifl given in Articles 66 to 68 of Byerly'a Fourier Series and Spherical Harmonics 
on: Ginn A Co., 1893). 



174 



JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



To use more than a few of the terms in Equation 5 for the evaluation of 
S/S » is very laborious because of the difficulty in evaluating ft, and F n . 
The number of terms required for a given degree of precision will depend 
somewhat upon the parameters B and y/b but is chiefly controlled by the 
parameter T. Computations show that very little error is introduced by 
neglecting all terms in the series except the first if T is more than about 
0.2; but many terms are needed for the usually desired precision if T is less 
than 0.02, — the smaller the value of T the greater the number of terms 
needed. Very precise values of S/S oo for small values of T may be found 
without the tedious computation indicated in Equation 5 by using 
another expression which will now be derived. 

Solution in terms of the probability integral As long as the shrinkage at 
the sealed surface remains negligible, the distribution of shrinkage from 
the exposed surface inward will be nearly independent of the distance 
between the two surfaces. Suppose that instead of considering the surface 
at y = to be sealed, the body is considered to be extended to infinity in a 

negative w-direction. Then instead of the boundarv condition — =0 at y 

dy 

= 0, the requirement will be 

S = (6) 

at y = — oo . 

The solution* satisfying Equations la, 2a and 6 and giving S = when 

t = Ois 



S_ 

Sc 



= 



[1-Jj 




[>-?] 


_2Vf . 


- 


_ 2 V T _ 



B(l 



|) + B*~T 



■ {') 



where <j> (x) is — = / e dx and T is again used in place of — . 



The quantity 1 — o ■ ■■ 



-4 f 



dx, is known as the proba- 



bility integral. Values of <t> [z) may be readily found by using a table of 
the probability integral. 

Numerical calculations show that Equation 7 gives values that differ 
from those given by Equation 5 by an amount less than the value of 
S/*Sa: at y =0; therefore, Equation 7 may be used in place of Equation 5 
whenever T i> so small that S Sm at y = is less than the permissible 
error. 



*Thi*> solution it very simiLar to that given for an analogous problem bv Carslaw in Article 25 of The 
Maciiiillan <k Co.. Ltd., 2d e<l., 1921). 



SHRINKAGE STRESSES IN CONCRETE 175 

Table 2 and Fig, 7, showing S/S « in terms of y/b, kt/b 2 , smdfb/k, were 
prepared from Equations 5 and 7. 

Stresses and strains 

Continuity, Hooke's law and equilibrium. As stated previously, the 
solutions for stresses are here restricted to homogeneous, isotropic solids 
that follow Hooke's law. Equations for the stresses that would be pro- 
duced in such a body by the shrinkages given by Equations 5 or 7 will 
now be derived. 

The shrinkage S has been denned as the linear unit deformation that 
would occur if each infinitesimal element were unrestrained. However, 
the properties of a continuous solid will not permit an arbitrary distribu- 
tion of deformations ; therefore, unless the distribution of shrinkage given 
by Equation 5 happens to be compatible with the conditions of continuity, 
stresses will be produced that will modify the deformations so as to make 
them compatible. Although in general six partial differential equations 
are required for a complete mathematical statement of the conditions of 
compatibility, 3 these are reduced to 

**- (8) 

dy 2 

for either long narrow beams (plane stress) or slabs (plane strain) if the 
stresses are considered to be independent of the longitudinal coordinate x. 
The term e x is defined as the resultant unit deformation in the z-diree- 
tion (the direction of length). It is therefore the algebraic sum of shrink- 
age, S, and the strain produced by stresses. a v is obviously zero; and if 
Poisson's ratio is zero or if the discussion is confined to narrow beams, a t 
is negligible. Therefore, 

e x = &-S (9) 

E 

or, solving for stress, 

<r, - E(e t + S) (10) 

where E is Young's modulus, 

The restriction imposed by Equation 8 requiring that the expression 
for longitudinal deformation contain no terms in y other than the first 
power (second derivative equal to zero) is equivalent to the assumption 
usually made in the elementary theory of beams that "plane cross-sections 
remain plane." If longitudinal restraint is complete, then e x is zero and it 
follows from Equation 10 that <r x = ES. If, however, longitudinal short- 
ening is permitted but complete restraint against bending is provided, 
then e x is not zero but is still independent of y. If the beam has no external 
restraint, the non-symmetrical distribution of shrinkage causes it to warp, 
making e x a linear function of //. For no external restraint the equations of 



176 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 

equilibrium (summation of forces in the j-direction equal to zero and 
summation of moments about the z-axis equal to zero) become 



/ 



<j z dy = 0. 



(11) 



and 



/ 



<*xV dy = 



(12) 



It may be shown by substituting Equation 10 into both Equations 11 and 
12 that if shrinkage (S) is either independent of or a linear function of y 
an unrestrained beam will be free of stress (e I = - S and c x = 0). For 
any other variation of shrinkage a stressed condition must result because 
the restriction on e x (Equation 8) will not permit it to be equal and oppo- 
site to S if shrinkage is a non-linear function of y. 

The only solution for e x that satisfies Equations 8, 10, 1 1 and 12 is 

b b 

= ( 6 v- 4 )i/ 5 ^ + ( 6 - 12 i)i /■**■<*> 



When this value of e x is substituted into Equation 10, the result is 



= E 



o o 



(14) 



Finally, 5 from Equation 5 may be substituted into Equation 14 thus 
g Tf, J 1 " 6 ! 8 /, 11 * n u *"-" ;'»""> »>eam as a function of the parameters 
V, o, kt/b,fb/k, 5 c and of Young's modulus. This substitution will not 
be made until later, because it seems advisable at this time to consider 
another approach. 

Solution by sup, rposition. Although the above derivation is short and 
is in the simplest form for checking the mathematical correctness of the 
equation, a derivation in which elementary solutions are superposed is also 
desirable because it will be easier in general to understand and because the 
hnal _ expresMon, are in more usable forms. In this second derivation the 
resultant stress ... U cowddered as consisting of three parts. The first part 
s that stress which would be produced by complete restraint against 
longrtudmal deformation; the second part is a uniform stress equal to and 
opposite m s,gn to the average of the first part; the third part is a stress 






SHRINKAGE STRESSES IN CONCRETE 



177 



resulting from a simple moment that is equal to and opposite in sign to 
the moment produced b\ r the sum of the first two parts. That is, the first 
part alone <j x f would result from complete restraint, the sum of the first 
and second parts <r/ would result from restraint against warping only; 
the sum of all three parts, i.e., a XJ would result if no external restraint were 
applied during shrinkage. 

Although in this derivation an expression for <r x appears to be the 
ultimate goal, expressions for a J and for a/ are also desirable. The stress 
trj may be representative of the stress in pavement slabs or building 
walls that are restrained from shortening and the stress <r/ is representa- 
tive of an unrestrained wall drying equally from two opposite sides (Case 
II discussed later). 

Since for complete longitudinal restraint e x = 0, it follows from Equa- 
tion 10 that the first part of the stress is 

aj = ES (15) 



i r 

Since the average value of trj is — I aj dy, the second part of 



b 

e r 

a x is — r- I S dy; therefore, the sum of the first and second parts 


(<r/) is given by 



a x " = E 



-\f 



Sdy 







• (16) 



The moment produced by <r/ is the moment necessary to prevent warp- 
ing. This moment per unit width of beam is found by multiplying Equa- 
tion 16 by y dy and integrating. This gives 



i- b 



M 







j *; ydy = E J Sy dy - -J I 



Sdy 



L 







.(17) 



For no external restraint this moment must be removed by superposing 
an equal and opposite moment. The stress resulting from a moment — M 
is given by the elementary theory of beams as 

M {y - 6/2) 
1/12 ¥ 



JOL. CONCRETE INSTITUTE January 1946 






T 1 

I / / 

L -I 



.•s N t!nT 

i 












■ , ., I V 









/ Ir 












Ofllhff'l 









- 












■ teaM 



SHRINKAGE STRESSES IN CONCRETE 



179 



in terms of the deflection iw it produces, the expressions for the stresses 
are put into more usable forms. When this is done, the following equations 
for stresses are obtained. 

For complete longitudinal restraint (first part of <r x ), 

a x > = ES (15) 

For restraint against warping only (sum of first and second parts of 

Oi 

<r x " = E (S - S av ) (20) 

For no external restraint (sum of all three parts of <r»), 

/ y\ 26^ 



• (21) 



Evaluation of the parameters ~ and — ^ 



When Equation 5 for 



S/Sc is substituted in Equation 18 for shortening and in Equation 19 
for warping and the indicated integrations are performed, the result is 

b °° 

1 






.(22) 



2b Vrri 



3l*S< 



i r s s av >y- 

= bj sZ ydy " 25. = 2- 



-rft 

e G. 



(23) 



where 



and 



H n = 



2B 2 



G, 



01 (B*- + B + fil) 



\cosB n 2 / 81 



If T t the non-dimensional time-factor, is small, the series in Equations 
22 and 23 converge rather slowly, and in that case it is convenient to use 
the following equations obtained by substituting Equation 7 into Equa- 
tions 18 and 19, respectively.* 



Sa 



e i <t>(B V T) - 1 + =- 



.(24) 



The lower integration limit for each integral is decreased from to — »; 



180 



JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



2bv„ 



WS< 



JL I_ 
2B + B* 



v* 



r>...(25) 



Furthermore, if the parameter B V T is very small, it is still better to 
use the following equations obtained by expanding the expressions in the 
brackets of Equations 24 and 25. 



^~ = BT 



4 _ 1 _ 

■= B V T +~(B V TY 



2bv n 



3Vt 
V BT 



15 V 



8 ,— 

(BVr) 3 + 



3Z 2 S, 



(24a) 



flr / b\/ 4 _ 



4 ^^i^v *■>•)- 



. (25a) 



In general the following rules will be found applicable for rapid evalua- 
tion of the parameters |= and ^t to a fair degree of accuracy. 

0<x> Ot^O m 

If T is more than about 0.05, use Equations 22 and 23. 

If T is less than about 0.05 and B is more than about 5, use Equations 
24 and 25. 

If T is less than about 0.05 and B is less than about 5, use Equations 
24a and 25a. 

Forces and moments necessary for complete restraint. The force neces- 
sary for longitudinal restraint is JV dA. Therefore, the average force 

b 
per unit area is j-J,,' dy. From Equations 15, 5, and 22 this becomes 




force per unit area = ES t 



2 



e II. 



■ (26) 



From Equations 17, 5, and 23, the moment per unit width necessary 
tor restraint against warping is found to be 



°° -Tfi 
U = £Sc & ^ e G n 

1 



(27) 



SHRINKAGE STRESSES IN CONCRETE 



181 



Simplification by taking B as equal to infinity. The principal equa- 
tions derived above reduce to simpler forms and the computation of 
numerical values is less tedious if the assumption is made that B, i.e., 
fb/k } equals infinity. If B is large, say 100 or more, the error introduced 
by assuming it to be infinity is negligible. However, if B is less than 
about 5, the error introduced by considering it to be infinity may be 
appreciable as is shown, for example, by Fig. 8, 9, and 14. Whether 
justifiable or not, the assumption that B = oo is frequently made in anal- 
ogous problems to which the diffusion equation applies. This assump- 
tion was made by Terzaghi and Frohlich 4 in developing the theory of 
settlement of foundations due to consolidation of underlying material, by 
Glover 5 in a study of distribution of temperature in concrete dams, and 
by Carlson 1 in a study of distribution of moisture and shrinkage in con- 
crete. The more important of the above equations for the special case 
of B = cd are given below : 

Equation 5 becomes 

S ^-4(-l)- 1 -(2n-l)*jT 

cos (2n - 1) — . 






s 



(2fl - l)ir 



2b' 



Equation 7 becomes 
Equation 22 becomes 



Sa 
Sc 



__8 s- 1_ 

" 7T 2 Z. (2n - 



- (2h - I) 2 (7r 2 /4) T 



I) 2 



Equation 24 becomes 
v -J T 

O oo \ 7T 

Equation 23 becomes 



2bv m 



WSc 



= z 



7T 2 (2n - \y 



i 



Equation 25 becomes 



2bv ma: 
3l 2 Sc 



yr . 



w (2n - 1) 



- 1 



_ (2n - I) 2 — T 
4 



T . 



182 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 

Tables, curves, and computations.* Tables and diagrams such as those 
by Newman 1 are available from which values of N S« and S m /Sd may 
be determined. However, such published tables arc in general not ade- 
quate for the present problem. The smallest value of the parameter T 
used by Newman in his computations was 0.1, whereas the stresses in 
cone sired for a much earlier period. The tables given 

here were prepared for T as low as 0.001. Moreover, so far as is known, 

er — ; not previously been evaluated fortius or any 

analogous problem and its bion is necessary for the problem here 

■ oi Equations 15, 20, and 21 for the theoreti- 
one side under the different modes of 

bh i luce quantities ■ , ' ; and - 

• ion ol ih« three parameters y b, B and 
I ig 7 Tin' second quan- 

Ild V i- given in Table 'A and shown 

ni:i!)til \ - '" B£ a function of B and 

• >/ ■ . s 

M\ in Fig 9. Tables . r > and 6 

20 and 21 as functions of i be 

prepared after the three 

1 abl< - 2, •'». and 1 1 Results foi 

0, li. L2, and 13 Fig. l i shows 

rsufi the parameb 
on <»i the tables and diagrams 

iaf ions based upon the 
i c used instead ol 1 lie 
mtered. I mple, 

q 7 then 

l ) ( u)i 

1.1/ 9.1 / 



• • »'•' ' " ■■•'■<■> ' .. .-.:... -...:.'.. | ,„••.! c i' » *-r :>!,!.► *.»♦- :,|, J COliM ;.U:\ 



SHRINKAGE STRESSES IN CONCRETE 183 

From mathematical tables 

i op; 
4> (1) = 0.15730; 4> (1.5) = 0.03389; e = 3.4903 

Therefore S/Sa> = 0.1573 - 0.03389 X 3.4903 = 0.0390. 

Note that this is the value given in Table 2 for the above values of 
B } T, and y/b. Also note that for the same B and T the table gives zero 
for y/b = 0, showing that it was permissible to use Equation 7 instead 
of Equation 5. 

When the theoretically correct equations are used, the computations 
are more involved. For instance, let T = 0.1 instead of 0.01 in the 
above example. T will then be so large that S/Sa> will have an appreci- 
able value at y/b = 0. Therefore, Equation 7 will not be applicable and 
Equation 5, the exact equation, must be used. A substitution of values 
for T and y/b into Equation 5 gives 

00 —QlQl 
A = i _ V F cos °- 8 ^ 

So* 2i ( " co$p n 

1 

The first step in evaluating the above expression is to determine n 
which Equation 2c shows to be a function of fb/k and n, i.e., B } and the 
integer n. The determination of |3 n by interpolation is simplified by the 
introduction of a„ where a n depends on B and n. The equation for 0„ is 
then written 

n = (n - 1 +a n )7r (28) 

Curves of a n versus B for the first six values of n and for n = 21 are 
shown in Fig. 2. By means of Fig. 2 and Equation 28 any desired j3 n 
may be found with reasonable accuracy for any value of B, The first six 
values of n for several different values of B are given in Table 1. 

After finding n for the given values of B and n, the factors F nt 
cosp n , cos (fin-) and e ~ T &* are determined. F n and cos /3 n as func- 
tions of B are shown in Fig. 3 and 4, respectively, for the first six values 
of n. The functions cos i & n -\ and e ~ l $ n are readily obtained from 

mathematical tables after the products 0« ~ and T &l have been deter- 

b 

mined. When the proper values of the four factors listed above are sub- 

S 
stituted, the above equation for^— becomes 

O as 



184 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 

0.84147 X 0.3152 X 0.4966 _ 0.1965 X 0.2161 X 0.9963 



O oo 



0.2541 
0.00844 X 0.1286 X 0.7280 
0.8101 



0.6277 



This reduces to — = 1 - 0.5183 - 0.0674 - 0.0009 = 0.4134 

The values for S/S » given in Table 2 were computed by one or the 
other of the methods illustrated above 

In preparing Table 3 from_which Fig. 8 was constructed (Fig. 8 shows 
shortening as a function of V T for various values of B), values of H n in 
Equation 22 were needed. Values of H n as a function of B for the first 
six values of n are shown in Fig. 5. In like manner, Fig. 6 showing G n 
as a function of B, served in the preparation of Table 4 from which Fig 
9 was constructed. Of course, for small values of T, Fig. 5 and 6 are not 
necessary since either Equation 24 or Equation 24a is used instead of 
Equation 22 and either Equation 25 or Equation 25a is used instead of 
Equation 23, depending on the value of B. 

Application to beams or slabs of any width-to-depth ratio when Poisson's 
ratio ts not zero. The effect of Poisson's ratio was neglected in the pre- 
ceding derivations. Its effect stated in general terms in the introductory 
remarks in regard to Case I will now be analvzed in more detail. If 
Poisson s ratio is not zero, Equation 9 for e, and Equation 10 for c x will 
be modified to include the effect of c t . That is, 

'*-w. = E(e. + S). (10a) 

However, if the ratio of width to depth is small, „ z will be negligible and 

■ tion 10a reduces to Equation 10. On the other hand, if the ratio of 

width to depth is very ] ar ge (a slab), the width being comparable with 

tlK-Je^th, then a. will be equal to „,. If „, = ,„ then Equation 10a 

°' = T~- (e * + V (10b) 

t J'Tn? 1 -' difference u between E^on 10b for a wide slab and Equa- 

.on 10 or a narrow beam is the factor * which occurs in Equation 10b 

bul ,,„ . Equation 10. Therefore, for stresses in a slab, E k ! Equations 

15, 20, and 21 is replaced by -£.. The stresses in beams „ dth- 

to-d..p,h nmo i, in„.r„K,lia, ( . will have stresses intermediate between 
of narrow beams and of slabs. Since ,o, appear in Equa- 



SHRINKAGE STRESSES IN CONCRETE 185 

tion 22 for average shrinkage nor in Equation 23 for warping, these 
quantities are the same for narrow beams and wide slabs. 

CASE II— SLAB OR BEAM DRYING FROM TWO OPPOSITE SURFACES 

Equations taken from those derived for Case I. Since the flow of mois- 
ture in a slab drying from only one surface is believed to be the same 
as that in either half of a slab of twice the depth drying from two op- 
posite surfaces, it will be assumed that the theoretical equations de- 
rived for shrinkage of a beam or slab drying from only one surface will 
apply equally well for either half of a beam or slab drying from two 
opposite surfaces. The plane midway between the drying surfaces will 
be taken as the plane y = as shown in Fig. 1 for Case II. Since the 
two halves of the beam will mutually restrain each other from warping, 
the equations for stresses, strains, and shortening in each half will be the 
same as those given previously in Case I for a beam restrained against 
warping and drying from one surface. 

CASE HI— RECTANGULAR PRISM DRYING FROM FOUR FACES 

Shrinkage 

The differential equation and boundary conditions. For a prism drying 
from four faces but not from the ends the diffusion equation reduces to 

\dy 2 dz 2 / dt 

The exposed faces of the prism will be taken as the planes y = =*= b, 
z = ± c, as shown in Fig. 1 for Case III. The boundary conditions then 
become 

^ = ±L(Sa> - S) (2a) 

dy k 

at the boundaries y = =*= b and 

3? = *£(fl. - S) (3b) 

dz k 

at the boundaries z = =*= c. 

The solution satisfying Equations lb, 2a, 3b, and giving S = at 
t = and S = S*> at t - °o is the following: 

-m cos A^ir" - r «* cosfij- 



O oo 



I 



COS f} n 
1 



Ye F m c 

cos p„, 



. (5a) 



1 

where p m F m} and T c correspond to p n , F n , and T, respectively, the differ- 
ence being that the dimension b has been replaced by c. 



186 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



Shrinkage expressed in terms of the solutions given for a prism drying 
from one face or two opposite faces. Since the infinite series in the first 
bracket in Equation 5a is identical with the one given in Equation 5 and 
the infinite series in the second bracket is like the first except that y is 
replaced by z, b by c, etc., and since Equation 5 applies to either half of 
a slab exposed on two opposite surfaces (Case II), it follows that the 
brackets have the following values: 



'. - - = 1 - & , 



2e 



■T e & 



cos f} n 

cos m - 
c 

cos fi m 



(29) 



= 1 - * 



.(30) 



where </> b is the value S/Sa> would have if only the surfaces y = ± b 
were exposed and <f> c is the value S/Sa> would have if only the surfaces 
z = =±= c were exposed. 

A substitution of Equations 29 and 30 into Equation 5a gives 



5— — 4>b + 4>c 

O 00 



<t>t>4>c- 



.(5b) 



Equation 5b shows that the evaluation of shrinkage for a prism dry- 
ing from four surfaces becomes a problem of adding the independent 
effects of drying from surfaces that are perpendicular to each other and 
then subtracting a term proportional to the product of the separate 
effects. 

For example, consider the shrinkage tendency at the point y = 0.46, 
2 = 0.8c in a prism for which c = 26 (width equal to twice the thickness)' 
Let/, k, and t be such that/6/fc = 5.0 and kt/b* = 0.20; then fc/k = 
10.0 and kt/c* = 0.05. </> 6 is found in Table 2 or from Fig. 7 to be 2398 
Since Table 2 was prepared for B equal to 0.1, 1.0, 5.0, and » only, and 
since fc/k = 10, 0, can be found from Table 2 only by interpolation 
However, examination of Table 2 indicates that for kl/c* equal to 05 
Equation 7 can be used instead of Equation 5 without appreciable error 
and therefore the equation rather than the table will be used to obtain 
<i> c - From Equation 7 



= | ' r | ~ * (■ °^— + 10 V0.05^ 

V2V0.05/ \2V0.05 / 



2+5 



.2V0.05/ V2V0.05 / 

From tables giving probability integrals and the exponential function 

4>c = 0.5273 - 0.000156 X 1097 = 0.355 
Therefore at y = 0.46, z = 0.8c, and t = 0.20 b-,'k 

s 

— = 0.2398 + 0.355 - 0.2398 X 0.355 = 0.510. 



SHRINKAGE STRESSES IN CONCRETE 187 

Shortening expressed in terms of the shortening of a prism drying from 
one face or from two opposite faces. The average shrinkage S at is given by 
b c 

s av = ~tff Sdy " : (18a) 


From Equations 5a and 18a 

-r# ro -T e 0L 

5fL_ = i - 5" e H n V e H m (22a) 

1 1 

or 

^- « H h + H c - HJI* (22b) 

O CO 

where H b is the value S S would have if only the surface y = ± b were 
exposed and // ifi the value S ar /'Sao would have if only the surfaces 
z = ± c were exposed*. Therefore, the average shrinkage, and conse- 
quently the shortening, if the bod ic, of a prism drying from four 
sides may be found by considering the separate effects of drying from 
opposite -ides in pa; 

For example, consider the shortening of the prism discussed above. 
H h is found in Table 3 to be 0.3510 and H c is found to be 0.1753. Therefore, 

— = 0.3510 + 0.1753 - 0.3510 X 0.1753 = 0.4- 7 

O QD 

Stresses and strains 

Nature of the problem and the method to be used to obtain a solution. In 
-■■s I and II previously discussed, where shrinkage was a function of 
time t and only one space coordinate y and where the problem was further 
simplified by neglecting the effects of the length and the width of the 
imen on the distribution of stresses, a solution for the one stress 
involved was readily obtained. However, in the problem now under con- 
sideration, a prism doing from four surfaces, shrinkage varies with an 
additional coordinate z. As a result stresses vary with this additional 
coordinate also, and more than one stress will be involved. The problem 
will be somewhat simplified by neglecting the effect of the length on the 
distribution of st e., the assumption will be made that stresses do 

not var>- along the length. The distribution of given by the 

solution based on this assumption will deviate a negligible amount from 

•If a prism were drying from all six surfaces, the corresponding equation would be SWS. = H. + Hh 

— He - IUH- - //-// f - H-H* — HaHiH,. Another way of expressing these relations is given by Glover 
VRdLI 



188 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 

the theoretically correct distribution when stresses in the central portion 
of a long prism are under consideration (principle of Saint Venant 8 ). 
The solution for shrinkage in terms of two space coordinates was air. 

when only one coordinate was involved (Equation 5a for 
I II compared with Equation 5 for Cases I and II j, because tendency 
to shrink is considered to be a scalar quantity. On the other hand, since 
stresses are tensor quantities, the solution for stresses usually becomes 
much more complicated whenever more than one coordinate is involved. 
In fact, elastieians have obtained exact solutions meeting all boundary 
conditions for only a relatively few problems in which stresses were func- 
tions of at least two coordinates and then only by considering the body 
to be infinite in the direction of one of these coordinates. The difficulty 
is that since stresses are tensors, boundary forces are vectors, and in 
two-dimensional problems two components of force must be satisfied at 
each boundary. The specified conditions of stress at anv two opposite 
boundaries can be satisfied by superposing particular solutions of the 
differential equations in accordance with the usual methods of Fourier 
analysis. But, in general, solution* ing rigorously the boundary 

requirements at two pairs of opposite boundaries simultaneously cannot 
be found by the usual metho 

A method of solving such problems after the appropriate differential 
equations have been derived was explained by the author in a recent 
paper*. That method will be used here. It is about the same as that u*ed 
previously by Taylor" and by Timoshenko" in analogous problems. 

Der ' the differential equations rehr , shrinkage Bv 

neglecting the variation oi and strains along the length of a body 

the problem becomes a two-dimensional problem in plane strain The 
flowing equations taken from the theory of elasticity are then applica- 

Equations of Equilibrium: 
<>y dz 

+ a^ =0 

Condition of Compatibility: 

Modified to Include Isotropic Shrinkage: 



dz 2 



' ~ E k ~ ^ 9 " ^ - S " - s - in ;hi ^ problem 



_ 



SHRINKAGE STRESSES IN CONCRETE 189 



€ v = — [— v.<*x + <r v - M°"z] " 


- 5 


e g = — [ — /x^r — mc v + ^] - 
E 


- s 


2 (1 + M ) 

7 U * = T M 





E 

The above seven equations giving relations between the eight unknown 
stresses and strains can be reduced to the following two equations by 
eliminating the four strains and the shear stress: 

<r M = M (<r ¥ + O +E(S- S av ) ■ (31) 

y »(,, + a: ) = JL_ V *S (32) 

1 - n 

where V 2 is written for — H . 

By* dy 2 

These two equations together with the two equations of equilibrium 
and the boundary conditions that 



fri 



and r yz = at y = ±6 



V «" 1,J - '&z 



cr z and T yz = at z = =*= c 
and Equation 5a for S constitute the mathematical statement of the 
problem. 

Solution for stresses. In general the stress a x will be larger than either 
ff v or ov The stress r vz will be relatively small in all cases. If only the 
value of the theoretical maximum stress is desired, a fairly good approxi- 
mation can be obtained by the following simplified formula: 

<r x = E (S - S av ) approx (31a) 

where S is given by Equation 5a and S av is given by Equation 22a. If, 
however, an accurate theoretical value of all stresses is desired, a complete 
solution must be obtained and this is given below: 

The solution for <r yt <r zy and r yz meeting all of the above requirements is 
as follows:* 

r 2 _ q 2 2 b ^- *r~ An 

*»= A °-lzt+~c Z Z -f^sa iy cosy ]Z 

t=i i=i 

♦Equations 33. 34. and 35 for stresses and Equations 36 and 37 for the coefficients B, and Ci given here are 
alnWt the same as Equations i, 2, and 3 for stresses and Equations 6 and 7 for the coefficients B n and A n 
MW*y4?&ento&L%™ The Aperies and the Antenna enter into the equations, given here in place 
of the S™?n the stress S given there; otherwise, except for slight differences in notation, the correspond, 
in^ Pauftfon^ are identical and the equations given here may be derived by the procedures given there. 

Equation 38 for X? and Equation 39 for A may be verified by substituting Equations 33 34, and 5a ». 
t Equa^on32andth e np^^ resulting equation within the domain 

under consideration by the usual Fourier analysis. 



190 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



, \ r, cos yjZ r • . / 

2- B >' T^Th ™ y SUlh ™ ~ ( 1+ y* b coth y& cosh y 3 y] 



i=i 



cosh y } b 



Z r cos dig r 
cosh ^ a%Z + ^~ aiC €oth a < c ) cosh Oiz] . . (33) 



, & 2 -3^ 



00 oo 



,_i, l^ + i Z Z 7T C0S ^ 



cos 7 ; z 



»*=1 3 = 1 



Z„ COS 7,2 r . , . 

'' c^sT^i biy smh ™ + (1 - T ' 6 co ^ ^ 6 ) cos * r,y] 
i=i 



\ r cos a f y r . , 

Z 'cosh af [aiZ Slnh aiZ ~ C 1 + a ' c co ^ «.<) «wA a, 2 ] 



(34) 



t'=l 

oo oo 



> — r sin any sin y,z 

00 

4- 'S" R Sin y > 2 I l , 

Z *' cos* 7i 6 ™ 6 c<rfA ^' 6 s '^ *» - 7,2/ cosA 7/ y] 

i=i 

00 

Z Ci cosA a,c LttiC C0 ^ a ' f s ''' i/! a ' z ~ «* cosh afi] . 



1=1 



■ (35) 



where 



a,- = 



iw 



b ' 



T/-2 



_ _c_ AoJ-iy , 6 

6 7T 



F+}I«t-z<->^ 



te«A l -= 



B,= 



i'=l 



i= 1 



i^io 



i + 



*(< 



m 



c, 



c 



) 



(36) 



SHRINKAGE STRESSES IN CONCRETE 



191 



c A.(-iy c 



(-iy ~ - y (_i) 

1" £■ — 



c, = 



i=i 






tan/i- — 



i=i 



i + 



(S)' 



B< 



6 V 



h^fcott™ 



to llll 






(37) 



.1., = 



" = 1 '" = 1 v"*«") v "ft"/ 

n = 1 m = 1 

The above equations, together with Equation 31 for a SJ constitute a 
complete solution on the basis of the given assumptions. 

In general, if both the parameters T and T c are equal to or greater than 
0.1, the series given above converge very rapidly so that only a few terms 
need be taken for a good approximation. The example given below will 
demonstrate the use of the above equations. 

Example: Stress at the middle of one side of a square prism for which B 
equals 5 and at a time for which T equals 0.1. If only one term in each series 
is used, the following values are obtained: From Equations 38 and 39 
and Table 1. 

ES " (0t + 0i)e -O-lfl.-O- 1 * // 



A = 
A. 



1-M 

ES c 



(1.3138* + 1.3138*) 0.8415* X 0.9130* 



.4. = - 2.0377 



i<:s 

1- 



ES<» (1.3138* + 1.3138*) 0.8415 2 X 0.9130 2 

An — ~ — — " — : — / _? \2 



(1 - *0 ** 



(l - _£-)' 

V 1.3138V 



192 

A n = - 0.0093 



JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 
E a oo 



l-M 



When these values are substituted into Equation 36, the result is 
(_ 2 _^ +0.0093) ^-'«5*£ Cl 



B l = 

1 + 7T (cotf* 7r — /an/i 7r) 

or since Ci = B Y (square prism), 

C, = B,= ~ 0.1472 — 

When the above values and z = c, y = Q are substituted into Equation 
33, the result is 

+ 0.0093 - 0.1472 X 0.3583 + 0.1472 X 0.97731 ^* 

Jl- M 



= p.0377 

L 6 



2 =C 

y = 



■[ 



+ 0.3396 + 0.0093 - 0.0527 



-r 0.1439 - 



■ = 0.4401 « 



z = c 
2/ = 



1-M 



When the summation of each series is carried to two terms, the result is 



ES 



0,4214 - — — and when the summation is carried to four terms in each 



1- 



ES : 



series the result is 0.4221 t^Jfl f or this stress. 

1- M 

The above shows that the series converge very rapidly for this example. 

The stresses a t and r vz are obviously zero at the point under considera- 
tion. 

From Equations 31, 5b, and 22b, Tables 2 and 3, and the above result 
for c y , 

= [j-j- 0.4221 + (0.0221 + 0.6913 - 0.0221 X 0.6913) 

- (0.2186 + 0.2186 - 0.2186 X 0.2186)1 £S«> 



z = c 
y = 



Li - M 



0.4221 + 0.3087 ES 



] 



2 = C 

y = o 



For all values of Poisson's ratio M less than 0.212, the above expressions 
will give larger values for <r y than for *„ This fact is of interest because, 









5E STRESSES IN CONCRETE 193 

in |bi - 

- 

OTHER METHOOS 

D tO 

- 

■ 
I 

M 

SUMMA- 

I 

ng, warp- 

I 
r beam & na one ^ ntaina an 

| n the correspond!!* 






194 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 

equation for a slab or beam drying from two opposite surfaces. Com- 
putations show that the equation with the added term gives much less 
stress tor the same size of body and the same period of drying. Compare / 

Tables 6 and 5. " ' 

Equations are given for all the stresses in a prism drying from all four 
surfaces, the third condition treated. These equations, though rather 
complicated in appearance, can be readily evaluated if the desired accur- 
acy is such that only a few terms in each series need be considered. For a 
rather rough approximation of the stress that is usuallv the most impor- 
tant in this third condition, the comparatively simple equation <r, = 
E(S ~ Scv) is recommended. 

Tables and curves are given from which the theoretical shrinkages, 
stresses, etc, may be obtained, at any point in the specimen after any 
period of drying, for various values of the physical properties, diffusivity 
surface factor, ultimate shrinkage, and dimensions of the specimen. 

Examples are given showing how numerical values mav be computed 
from the equations and how the tables and curves may be used. 

REFERENCES 
JaLFe?. m g 7 S &\ ?e ^,^T27 COI1Crete MeB ^ n '" »* * "■ Carlson, ACI Jolr.val, 

^^^'^^^^^^trst^hk and E - s - Sokolnikoff ' 

3. Theory of Elasticity, by S. Timoshenko, McGraw-Hill. New York, 1934, p. 196 
Fran, S^^S^^St^" * * * *«*» "* °. K ' ™** 

<rfCtenri^a?"'^ T ^ e GIo^ D ArT i T U,i0n in a v SucC ^ ion 0{Ulis Due t0 ReI ^ 
« "TkrT * „ OIo%er ' ACIJovR * A L. ^v.-Dec. 1937,Proc.. V.34, p 105 

In£. 0f T C h heS O Enl., i^ P g Tl9 S 31 lidS ' , ' b> ' * * Xwman ' Interim Publication, Am! 

"Flaw of Heat in DanV' by R. E. Glover, ACI Journal. Xov,Dee. 1934, Proc. 

8. Reference (3), p. 31. 

pp 222^" ° fPtaUS ^ SMh - ** S - Timoshenko, McGraw-Hill, Xew York, 19M,' 
12. Ref. (3), pp. 21-23. 

C^^bfati^p t JX%^^S^S!' T^T-? 1 fi*— » Concrete 
14 "A «JLi ArTi j , 1 ^ r ° c ^ m - • ,Jf Testing Matenals, V. 39, 1939, p. 913. 

ss?" "*£& ass ra/raaarafstat 



SHRINKAGE STRESSES IN CONCRETE 195 

17. Relaxation Methods in Engineering Science, by R. V. Southwell, Oxford Univ. 
Press, 1940. 

18. "A Lattice Analogy for the Solution of Stress Problems," by Douglas McHenry, 
J. Inst, of Civil Engrs., Dec. 1943, pp. 59-82. 

19. "Numerical and Graphical Method of Solving Two- Dimensional Stress Prob- 
lems," by H. Poritsky, H. D. Snively, and C. R. Wylie, Jr., J. Applied Mechanics, 
June 1939, V. 6, No. 2, p. A-63. 

20 "The Effect of Change in Moisture-Content on the Creep of Concrete under a 
Sustained Load," by G. Pickett, ACI Journal, Feb. 1942, Proc. V. 38, p. 333. 

21. "Die Beeinflussung des Schwindens von Portlandzement durch Sulfate," by 
G. Haegermann, Zement 28 (40) 599 (1939). 

22. "A New Aspect of Creep in Concrete and Its Application to Design," by Douglas 
McHenry, Proc. Am. Soc. Testing Materials, V. 43, 1943, p. 1069. 

23. "Attempts to Measure the Cracking Tendency of Concrete," by R. W. Carlson, 
ACI Journal, June, 1940, Proc. V. 36, p. 533. 

24. "The Influence of Gypsum on the Hydration and Properties of Portland Cement 
Pastes," by W. Lerch, unpublished. 

25 "The Dependence of the Shrinkage of Portland Cement on Physical and Chem- 
ical Influences," by H. Kuhl and D. H. Lu, Tonind. Z. 59 (70) 843 (1935). 



196 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



m 






• 7T~" 



3_ 




Fig, 1 — Illustrations of the con- 
ditions treated. Shading indi- 
cates sealed surfaces. Case I 
— Beam (or slab) drying from 
one face only. Case II — Beam 
(or slab) drying from two op- 
posite faces. Case III — Prism 
drying from four faces. The 
ends of the prism at x = * a 
are sealed. 



OSOp 
0.45 - 
040 - 
OiS - 
050 - 
n 0« - 
020 - 

o ig - 
010 - 
005 - / 

0-0 











»here (J n is the n?> root ^^ 




* P n ton(i n .fjL^ B 






i 




«y / 








y 


// 




X 


/ / v%/ 




^/ 


S SS; 








* 


— -^^^ 


= ss: ^--^^ 






B (loq let 
Fig. 2 — Curves for the determination of /3„ 




Fig. 3— Relationship between F, and H 



SHRINKAGE STRESSES IN CONCRETE 



197 




B (log scale) 

Fig. 4 — Relationship between cos 0„ and B 




B (log scale) 



Fig. 5 — Relationship between H n and B 



ato - 

0.09 - 
0.08 - 

ao? - 

o.ot> - 

n 

0.05 - 
0.04 - 
0.03 - 

o.oz 




B ( log scale 



Fig. 6 — Relationship between G n and B 



198 



JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



T T~ 


f- A- T 


IT \ $ . 


P > I" 


|\ \~ \\_ ' 


|\K\ \ 


^^^$\ V- 


l^^^^^sA 


1 S? S 2 S g $ g S 2S° 



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Bj 

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"J 

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SHRINKAGE STRESSES IN CONCRETE 



199 

















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200 



JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 



||5* 







. 



Fig. 14-Maximum stress (maximum value of stress at exposed surface) and 
maximum warping vs. the parameter B 



TABLE 1-0„ 
n tan n = B 



B 

0.0 
0.1 
0.5 
1.0 
2.0 
5.0 
10.0 
100.0 





0.311053 

0.653271 

0.860334 

1.076874 

1.31384 

1.42886 

1.55525 

7T 2 



3.1731 
3.2923 
3.4250 
3.6436 
4.0338 
4.3058 
4.6656 
3tt 2 



ft 



2tt 

6.2991 
6.3615 
6.4372 
6.5783 
6.9097 
7.2281 
7.7760 
5tt/2 



0% 



3tt 


4, 


5w 


9.4354 


12.574 


15.715 


9.4774 


12.606 


15.740 


9.5292 


12.645 


15.771 


9.6296 


12.722 


15.834 


9.8927 


12.935 


16.010 


10.200 


13.213 


16.260 


10.887 


13.998 


17.109 


7tt 2 


9*72 


llir/2 



_^ 



N CONCRETE 



201 



TABLE 2— RATIO OF SHRINKAGE (OR SWELLING) TO ULTIMATE SHRINKAGE 

(OR SWELLING) 









/ 




















































' 


















































































































I 






































































• 






































































' 



































































































































































































































I JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 

TABLE 3-UNIT SHORTENING-AVERAGE UNIT SHRINKAGE (|= ) 



1 v ~ T ^u 

= 1 - 2 e H„ 






0.1 



B = 0.5 



B = 1.0 



2.0 



5.0 



B = 10.0 



B = 



0.001 

0.002 

0.003 

0.005 

0.010 

0.015 

0.02 

0.03 

0.04 

0.05 

0.075 

0.10 

0.15 

0.20 

0.30 

0.4(1 

0.50 

0.75 

1.0 

1.5 

2.0 

3.0 

4.0 

5.0 

7.5 

10.0 

15.0 

20.0 



O IIOIIl 

0.0002 
0.0003 
0.0005 

O OIIK) 

0.0015 

0.0020 

0.0030 

0.0039 

0.0049 

0.0074 

0.0098 

0.0146 

0.0193 

0.0288 

0.0382 

0.0474 

. 0702 

0.0924 

0.1353 

0.1761 

0.2521 

0.3211 

0.3837 

O .-.161 

0.6301 

0.7658 

B556 



0.0005 

0.0010 

0.0015 

0.0024 

0.0048 

0.0072 

00U5 

0.0141 

0.0185 

0.0231 

0.0340 

O 0446 

0.0653 

0.0854 

0.1239 

0.1606 

n 1057 

0.2771 

0.3502 

0.4751 

0.5767 

0.7233 

0.8182 

0.8821 

0.0594 

0.9860 

i) 9983 

i» gggg 



O 0010 

0.0019 

0.0029 

0.0047 

o 009:1 

0.0137 

0.0181 

0.0265 

0.0347 

0.0426 

0.0620 

0.0803 

0.1154 

0.1489 

0.2064 

. 2666 

0.3188 

0.4339 

0.5296 

0.6751 

o 7756 

0.8930 

(I 04 SM 

. 9756 

0062 
. 9994 

1 oooo 



0.0019 

0.0037 

0.0055 

0.0090 

0.0173 

0.0252 

0.0317 

0.0473 

0.0611 

0.0742 

. 1050 

(I 1336 

. 1860 

0.2327 

0.3190 

0.3939 

4604 

II 5062 

O 0970 

. 8306 

. 9052 

O 0625 

O 0907 

0.9971 

0.9998 

1 . 0000 



0.0045 

0.0085 

0.0124 

0.0196 

0.0370 

0.0515 

. 0649 

0.0891 

0.1115 

0.1319 

0.1779 

0.2186 

(i 2895 

0.3510 

0.4555 

0.5422 

0.6148 

0.7498 

0.8375 

0.9315 

0.9711 

. 9949 

0091 

it 9996 
1.0000 



0.0080 

0.0147 

0.0206 

0.0323 

0.0556 

. 0755 

. 0932 

0.1242 

0.1512 

0.1753 

. 2282 

0.2739 

0.3510 

0.4167 

0.5258 

0.6136 

. 6849 

0.8109 

0.8865 

0.9591 

0.9857 

0.9981 

0.9998 

1.0000 



0.0357 

0506 

0.0619 

0.0800 

0.1129 

. 1383 

0.1596 

0.1954 

0.2257 

0.2523 

0.3090 

3506 

. 4370 

504 1 

0.6133 

0.6979 

0.7641 

0.8726 

0.9313 

0.9800 

0.9942 

0.9995 

1.0000 



TABLE 4- WARPING (%~ in thousandths) 





0.0010 
0.0015 
0.0020 
0.0030 

0040 

0.0050 

0.0075 

0.010 

0.015 

n 020 

0.030 

(HO 

0.050 

0.075 

0.10 

0.15 

0.20 

0.30 

0.40 

o 50 

0.75 

1.0 

1.5 

2.0 

3.0 

4.0 

5.0 

7.5 

10.0 

15.0 

20.0 



B = 0.1 




(I 04S 
0.071 
0.093 
0.137 
0.180 
0.222 
0.324 
0.421 
0.606 
0.778 
1.092 
1.375 
1.628 
2.161 
2.568 
3.126 
3.457 
3.757 
3,843 
3.851 
3 782 
3.694 
3.519 
3.353 
3.044 
2.763 
2 50 s 
1.969 
1 506 
0.953 



26 IW 



2 

1 



-T$\ 



B = 0.5 B = 1.0 



2.0 



B = 5.0 





0.235 
0.348 

(I 15s 
-.74 

883 

1 (Ills 

1 579 

2 04 5 
2.920 
3.735 
5.415 
6.518 
7 s:i2 

10.138 
1 1 . S46 
14.017 
15.128 
15 773 
15 546 
15.043 
L3 588 
12 217 
o 869 
7 059 
5.203 
3.417 
2.216 
0.763 
0.262 
0.031 
004 




0.46 
0.68 
0.90 
1.32 
1.72 
2.12 
3.06 
3.94 

7.08 
9.74 
12 05 

14.03 
17.96 
20 75 
23.96 
25 29 
i:* 61 

24.14 
22.59 
is S5 
15.67 
10.82 
7 17 
3 . 57 
1.70 
0.81 
. 1 3 
02 
0.00 




0.91 
1.33 
1.73 
2.54 
3.31 
4.03 
5.74 
7.35 
10.25 
12 92 
17.34 
21.07 
24.21 
30 os 
33.87 
37.56 
38.32 
36.28 
32.88 
29 43 
22.07 
16.52 
9.25 
5 is 
2.05 
0.51 
0.16 
0.01 
0.00 




2.13 
3 os 
3.99 
5.73 
7.23 
8.69 
13 Os 
1 5 . 1 s 
20 4 5 
24.97 
32.63 
38.32 
42.70 
49.98 
53.81 
55.70 
53.89 
47.03 
39.90 
33.64 
21.86 
14.20 
5.09 
2.53 
0.45 
0.08 
0.01 
0.00 





3.82 
5.43 
S.89 
9.61 
12.03 
14.27 
19.16 
23.35 
30.38 
35.92 
44.71 
50.93 
55.33 
62.22 
65.16 
64.50 
60.56 
50.52 
41 .38 
33 77 
20.27 
12.17 
4.38 
1.58 
0.20 
0.03 
0.00 




16.83 
19.79 
23 . 22 
27.90 
31.66 
34.89 
41.36 
46 . 42 
54.12 

67.67 
72.80 

76.15 

79.76 
80.40 

74.18 
66.85 
52.74 
41.27 
32.25 
17.40 
9.39 
2.74 
0.80 
0.07 
0.01 
0.00 



, 



SHRINKAGE STRESSES IN CONCRETE 



203 



TABLE 5— RATIO OF STRESS a" T IN AN UNRESTRAINED BEAM DRYING FROM 

TWO OPPOSITE SIDES (OR IN A BEAM DRYING FROM ONLY ONE SIDE 

AND RESTRAINED AGAINST WARPING) TO THE ULTIMATE STRESS 

FOR COMPLETE RESTRAINT ES*> 






S 



Sa 

Si 





v 


y 


y 


y 


V 


y 


y 


V 


V 


y 


v 


V 


T 


-=0 


-=0.2 


-=0.4 


-=0.6 


-=0.8 


- = 1.0 


-=0 


-=0.2 


-=0.4 


-=0.6 


-=0.8 


-=1.0 




b 


b 


b 


b 


6 


6 


& 


b 


6 


b 


6 


b 




B=0.1 






B = 


1.0 






0.0 






































0.005 


-.0005 


-.0005 


-.0005 


-.0005 


-.0003 




-.0047 




-.0047 


-.0047 


-.0040 


.0703 


0.010 


-.0010 


-.0010 


-.0010 


-.0010 


-.(1001 




-.0093 


-.0093 


-.0093 


-.0091 


.0000 


.0942 




-.0O15 


-OKI 5 




-.0014 


,0006 


.0120 


-.0137 




-.0137 


-.0125 




.1101 


0.020 


-.0020 


-.0020 


-.0020 


-.0017 


.0013 


.0138 


-.0181 






-.0148 


1 24 


.1235 


0.030 


-.0030 




-.(11)20 


0019 




2 


-.0265 


-.020 1 




-.0170 


.0246 


.1425 


0.(11(1 


-.0030 


-.0030 




-.0019 


.0041 


.01 s.3 


-.0346 


-.0344 


;| 1 


-.0165 




,1563 


0.060 


-.0049 


-.00 is 




-.0019 


.0051 


.0198 


-.0123 


-nil', 


-.0361 


-.0152 


<<l I 1 


.1670 


0.075 


-.0072 


-.0000 




-.001 » 


.0072 


.0228 




-.0570 


-.0112 


-.0104 


0616 


.1851 


0.10 


-.0090 


-.OOSl 


-.1.110 1 




.Hi .on 






-.0682 




-.0054 




.1961 


0.15 


-.0117 


-.0106 


-.0069 


.oooo 


.011 1 


.0277 


-..010 


-.0821 




.001 1 


.0871 


.2064 


0.20 


-.0133 


- 0119 


-.0074 


.0005 




.0293 


-oooo 


-.0880 


-.0539 




.0929 


.2081 


0.30 


-.0149 


-.0132 


-.0079 


.0010 


.013S 


.0306 


-.1028 


-.0900 


-.0530 


,0089 


.0948 


.2027 


0.40 


-.0154 


-.0136 


-.OOS] 


.0011 


.0141 


,0308 




-.0855 


-.0497 


01)0 1 




.1893 


i >..-,() 


-.0154 




-.0080 


0013 


.0142 


.0311 


-.001 1 


-.0S01 




.0088 




.1766 


0.75 


-.0152 


-.0133 


-.0079 


.0013 




.0302 


-.0764 


-.0669 




.0074 


.0699 


.1471 


1.0 


-.0148 


-.0130 


-.0(i77 


.0012 




.0295 


-.0634 


-.0555 


-.0321 


.0062 


.0582 


.1222 


1.5 


-.0140 


-.0123 


-.0072 


.0011 


.0131 


,0281 


-.0439 


-.0384 


-.0223 




.0401 


.0844 


2.0 


-.0134 


-.0118 


-.0070 


.0011 


.0124 


.0267 




-.0266 


-.0153 


.0029 


.0277 


.0583 


3.0 


-.0 122 


-.0107 


-.0003 


.0010 


.0113 


.0243 


-.0145 


-.0127 


-.0074 


0013 


.0131 


.0277 


4.0 


-.0111 


-.0007 


-.0057 




.0102 


.0221 


-.0069 


-.0060 


-.0035 


,0007 




.0133 


5.0 


-.0101 


-.on.v.t 


-.0052 


9 


0092 


.0200 


-.0032 


-.0028 


-.0016 


.0001 


.0030 


(.004 


7.5 


-.0070 


-.0069 


-noil 


0006 


.0073 


.0157 














ion 


-.006] 


-.0055 


-.0033 




.0055 


.0120 














15.0 


-.0038 


-.0033 


-.0020 




.0035 


,0076 














20.0 


-.0023 


-.0021 


-.0012 


.0002 


.0022 


.0047 




















B = 


5.0 






£ = «> 


0.0 






































0.001 


-.00 1". 


-.00 15 




-.0045 


-.0045 


.1514 




-.1.357 


-.0357 


-.0357 


-.0357 


.9743 


0.002 


-.OOS5 


-.0OS5 


-.0085 


-.0085 


-.0084 


.2010 


-.0506 


-.0506 








.9494 


0.0(15 


-.0100 


-.0196 


-.0100 




-.0124 


.2823 


-.0800 


-.0800 






-.0345 


.9200 


0.010 


-.0370 


-.0370 


-.0370 




.0020 




-.1129 


-. 1 1 20 


- L127 




.0412 


.8871 


0.015 


-.0515 




-.0515 








-.1383 


-.1383 


-.1378 


-.1171 


.1100 


.8617 


0.020 


-.0649 


-.0646 




-.0523 




.4120 


-.1596 


-.1595 




-.1141 


.1577 


3404 


0.030 


-.0890 


-.Os.sl 


-.Osls 


-.0538 


.0871 


.4441 




-.1944 








.8046 


0.040 


-.1 100 


-.1098 


-.0998 


-.0481 


.117., 


.1000 


-.2249 


-.2210 


-,10 1s 


-.0684 


.2538 


.7713 


0.050 


-.12! tO 


-.1277 


-.1001 


- 0401 


.1404 


.4707 


-.2492 


-.2408 


-.1945 


-.0464 


.2748 


.7477 


0.075 


-.17(10 


-.1604 


-.1207 


-.0200 


.1765 


.4778 


-.2892 


-.2683 




-.0073 






0.10 


- 1965 


-.1700 


-. L230 


-.0044 


.1948 


.4727 


-.3096 




-.1771 


.0151 






it. 15 


- 2160 


-.1969 


-.llso 


-.0130 


.2052 


t hi 


-.3012 






.0317 






1 ». _'( 1 


-.2164 


-.ISO 7 


-.1112 


-.0199 






-.2765 


-.2404 








.4959 


0.30 


-.lols 


-.H.71 


-.0948 


.0216 


.17 10 


.3551 


-.2200 


-.1000 


-.1048 


.0292 


.1986 


.3867 


0.40 


-.1634 


- 1 1J-' 


-.OSOI 


.OI01 


.1485 




-.1723 


-.1492 


-.0818 


.0231 




,3021 


0.50 


-.1370 


-.1 103 


-.01,7 1 


0103 






-.1347 


-.1166 


-.0640 


.0180 




.2359 


(J. 7.'. 


- osoo 


-.0770 


-.0437 


0106 






-.0727 


-.0629 






.0655 




1.0 


-.0582 


-.0506 


-.Ojsj 






.1064 


-.0393 


-.0341 


-.01S7 




.0353 


.0687 




-.0246 


-.0214 


- 0170 




.0223 


.0448 


-.0114 


-.0100 




.0015 




.0200 


2^0 


-.0104 




-.0051 


. (HII2 


.0094 




-.0034 


-.0029 


-.0016 


.0004 


.0030 


.0058 




-.0019 


-.0016 


-.0009 




.0016 

















204 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 

TABLE 6— RATIO OF STRESS <r, IN AN UNRESTRAINED BEAM DRYING FPniu 
ONLY ONE SIDE TO ULTIMATE STRESS FOR COMPLETE RESTRAINT g? 



0.0 
0.005 

0.010 
0.015 

0.10 
0.30 

1.5 



0.001 

- 






-f= + («- 12*) 

Ox s 



2bv. 






v v y 

-=0.2 -=0.4 -=0.6 
b b b 



v y 

= 1.0 
6 6 



B = 0. 1 



y „ y y y y y 

-=0 -=0.2 -=0.4 -=0.6 -=0.8 - = 1.0 
b b b b b b 





.0021 

.0071 



.0003 
.0005 

9 

.0011 

.0011 

.0003 

.0003 

.0001 
.0001 

.0001 
.0000 





- 

-.0019 
28 

-.0017 

-.0014 





-.0014 
-.0005 





-.0016 
-.0015 

- 

.0001 

.0003 

.0001 
.0001 




78 

.0100 

n 

- 
- 

.0060 

.0030 

- 



B 



1.0 




0080 

.0198 
.0343 

.o;uo 

.0419 

.0510 

.0473 
0441 

.0210 
.0145 




.0029 
.0049 

.0074 
.0087 

.VO\'0 

.0089 

.0042 
.0030 
.0022 

.0014 

- 

.0010 

.0001 




-.0022 
-.0046 
-.0070 
-.0091 
-.0135 
-.0169 
-.0193 
-.0226 
-.0237 
-.0243 

22 

- . ' 

-.0102 

-.0160 

-.0133 

- 

- 

-.0031 

-.0015 

-.0006 




-.0072 
-.0138 
-.0192 

-.0233 

-.0310 
-.0320 
-.0320 
-.0303 
-.0274 
-.024 7 

-.0218 

-.0199 
-.0183 

-.0132 
-.0120 

-.0061 
-.0030 
-.0013 
-.0006 




-.0116 
-.0142 
-.0142 
-.0131 
-.0106 
-.0081 
-.0061 
-.0031 
-.0012 
.0008 
.0019 
.0026 
.0025 

.0020 
.001b 
.0011 

.0002 
.0002 

.0001 




.0576 
.0706 
.0766 
.0811 
.0641 
.0840 

.0773 
.0716 
.0626 

.0490 
.0445 
.0318 
.0340 
.0282 
.0195 
.0135 
.0063 
.0031 
.0015 



B = 5.0 






.0141 

-- 
.0011 

- 

.0001 
.0000 





- 

5 

- 
-.0004 



- 
- 



28$ 

3 

27 

7 
.0000 





" 

.1142 

72 

.0006 



B = 



887 

- 
728 

>8S 
.0014 





.0411 

.0105 
.0027 
.0001 
-.0009 

-.0006 

-.0002 
.0002 

.0000 
.0000 





-.1044 
-.1031 

-.0641 

-.0416 

-.0021 

,0006 





- 781 

-.121b 
-.1626 

-.1*23 

-.1741 

78 
-.1031 
-.0814 

57 
-.0459 
-.0340 

-.0111 

-.0018 

-.0006 





-.0963 
-.1326 
-.1601 

-.00b3 
.0005 

.0093 

.0117 
.0119 
.0112 

.006S» 
.0053 
.0029 

.0015 
.0004 
.0001 





.8101 
.7107 

.5370 
.4817 

.3375 
.2905 
.2122 
.1670 
.1179 

.0545 

.0010 



Title 42-8 — a part of PROCEEDINGS, AMERICAN CONCRETE INSTITUTE Vol. 42 

JOURNAL 

of the 

AMERICAN CONCRETE INSTITUTE 

(copyrighted) 
Vol. 17 No. 4 7400 SECOND BOULEVARD, DETROIT 2, MICHIGAN February 1946 



Shrinkage Stresses in Concrete* 

By GERALD PICKETTf 

Member American Concrete Institute 

PART 2-APPLICATION OF THE THEORY PRESENTED IN PART 1 
TO EXPERIMENTAL RESULTS 

Carlson's results on prisms drying from one end 

As mentioned in Part 1, Carlson 1 applied diffusion principles to the 
problem of computing both loss of moisture and distribution of shrinkage. 
The fundamental equations on which his computations were based are 
the equations to which Equations 5 and 22 of Part 1 reduce when the 
parameter B is set equal to infinity. In his experimental work the 
prisms were allowed to dry through one end only, the rest of the surface 
being sealed. Measurements were made over gage lines that were 
parallel to the direction of flow of moisture. These conditions appear to 
be most favorable for the direct measurement of the distribution of 
shrinkage tendency since in an unrestrained specimen shrinkage stresses 
should not have any appreciable effect on the unit shortening in the 
direction of moisture flow. 

In Fig. 3 of his paper Carlson showed two diagrams. One diagram 
gave the distribution of shrinkage as measured after a definite period of 
drying and the other gave the computed "distribution of drying" (loss of 
moisture) for different assumed coefficients of diffusion for the m 
period of drying. The observed distribution of shrinkage and the com- 
puted "distribution of drying" are in good agreement when the proper 
coefficient is selected. However, as shown by Fig. 1 of his paper, the 
measured loss in weight was not in very good agreement with the theory. 
Carlson could have obtained slightly better agreement between theory 
and measured shrinkage if he had taken surface conditions into account, 

♦Part 1 of this paper was published in the ACI Journal, January. 1946, and includes (p. 194) the com- 

Pl VPr;5e^or e o f f Applied Mechanics. Kansas State College, Manhattan, Kan., formerly Portland Cement 
ition Research Laboratory, Chicago. 

(361) 



362 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 




--rS - ^> 

■ - -- - .-« - <y 

_\ - f5 _— r ---_--r-.-_- : m =zaoox/a1 

f=a07/n./cs - oerd&y 

- -£ -*r - cemeni 

stfiaa - _■ - f 



Fig. 1 5— Comporiscn of observed and calculated course of shrinkage 

i.e., used a finite value for the parameter B. However, had he done so, 
the discrepancy between theory and measured loss in weight would have 
been greater than that shown. 

Carlson's work is important evidence in support of the hypothesis 
that shrinkage of concrete approximately follows the laws of diffusion. 

Shrinkage of prisms of various sizes drying from one or more sides 

In the work done in this laboratory measurements were made on 
gage lengths transverse to the direction of moisture flow. Since variations 
in shrinkage along the path of moisture flow result in stresses transverse 
to the direction of flow, the measurements include the strains produced 
h J These however, the specimens are long compared to their 

dimension m the direction of moisture flow and the stress-strain relation 
is linear, then as shown in Part I the shortening of the central axi< will 
be equal to the average shrinkage. The results to be discussed provide 
a test of the theory for conditions in which both size of specimen and 
number i j es are variable. 

the unit shortening versus davs of drving for three 
different sizes of prism* of the same mix and for the three different 

drying conditions discussed in Part 1. .Mix A and cement M, described 
in the Appendix, were used. The specimens were cured seven davs 
under water. Each point is the average of the results from two pn~ 
lne curves were constructed from computations based on the theoretical 
equations developed in Part 1. These equations, which give the theoreti- 



SHRINKAGE STRESSES IN CONCRETE 



363 



unit shortening = S a 



cal relationship between unit shortening, the constants of the material, 
and dimensions of the specimen have the form 

\ k ¥ bf 

where S a> is ultimate shrinkage for the assumed final drying, / is the 
surface factor, k is the diffusivity factor, and b and c are dimensions 
of the specimen. The exact form of the function, especially the way in 
which c/b enters into it, differs with the drying condition. 

The three constants £«,, /, and k were evaluated from average ex- 
perimental values for the pair of prisms of 2-in. square cross-section, 
drying from four exposed sides. From these same constants the curves 
were constructed, as shown in Fig. 15, not only for this pair but also for 
the theoretical unit shortening of the other eight pairs of prisms. The 
agreement between the experimental values and the calculated curves is 
fairly satisfactory except for two pairs of 3x3-in. specimens, which were 
observed to have cracked during drying and therefore could not be 
expected to shorten in accordance with the theory. 

Discussion of the validity of the theory on the basis of the foregoing data 

The data from those specimens that did not crack, together with 
the data given by Carlson, might seem to indicate rather conclusively that 
shrinkage does take place in accordance with the theory develops I in 
Part 1. However, such a conclusion would not be justified. A good fit 
between an equation and experimental data is necessary but it is not 
sufficient proof of a theory. Although constants in the equations of 
Part 1 may be chosen so that the theory given there will be in good agree- 
ment with experiment for certain measurements on spc.imens under a 
few different conditions, the theory should be expected to fail under 
some other conditions since it rests on some assumptions that are not 
wholly correct. Shrinkage is not linearly related to change in moisture 
content; the flow of moisture in concrete does not follow the law of 
diffusion; and the stress-strain relation is not linear. Since the assump- 
tions are not wholly correct, the factors S «> , /, k, and E that are supposed 
to characterize the material must be empirical, and experimentally 
determined numerical values of these factors will be different for different 
Tests on the same material. The good agreement between theory and 
the experimentally determined contraction of the specimens discussed 
above must be the result of the balancing of opposing effects. They u ill 
not necessarily balance the same way in another test. 

The foregoing criticism means that however promising the theory 
may appear from the results of a few experiments, the application of the 
theorv must be limited and extrapolation of the results to sizes of speci- 
mens or conditions of drying other than those for which the constants 



364 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

were determined cannot be made with confidence. The selection of 
•r given conditions constitutes the chief difficulty for the 
prad of the equations. This doee not mean that the theory is 

of little I considerable value. 

Tnn l equations are not rigorously correct and the 

*ants cai the meaning attached to them. But the 

linkage does follow the diffusion equation approxi- 
mately and that the defn a are approximately those 
gi the theoretical equations if the empirical constants selected 
to give fair agreement with experimental result-. 
Although the experimental results shown in Fig. 15 appear to be 
in good agreement with the theory, a close study shows the following in 
regard to those pri~ og from only one side: I 1 After pro 

-i of the prisms drying from one fflde I 
- • ^ indicated by the theoretical curv< 

- --:u. prisms deviate more from 

•' Lxl-in The lxl-in. 

the same lvalues as the 2x2-in. 

-till more I 
- -- - tion. I- 

Lid result in still greater di>- 

omputed values unless allow] 
th change in lvalue. 

Comparison of dota on warping with data on shortening 

ded by the results shown in 

•.re root eriod 

! the prism.* As mentioned in 

is The abscises give* a nearly straight 
i entering and warping. 
■ the thickness of the specimen puts 
able basis. Multiplying the 
-- span by the thickness put* the 
on the same basis in regard to unit defoi 
J j l 
- 15 that were drying from only 
- 

trying fn 
- 

OS. 
iiammc* in the dmiio: kn-iu 



SHRINKAGE STRESSES IN CONCRETE 



365 



4000 



V) 




JZ 




-t— 




c 




o 






V)00 


b 




c 




en 

c 


2000 


c 




0) 








I— 




o 




x: 


1000 


-»- 




c 




3 









<^ 




in 


.40 


<U 


c 


r 







& 




S.30 




r\, 


v> 


*) 


<L> 




E 


t) 






t— 


£.20 


c 


h 


o 


<ij 


^ 


< 


u 


tj 


dj 


.C: 






Q) 


*.10 


o 


Ci 


t— 


a 


OJ 


0- 




h 


f 




QJ 


* 


CJ 



Points are Experimental I 
Curves are Theoretical, based on 

/* - o.o7-in. per day 

— k -a 0J5 -/a ? p £T flto^" 




12 
10 

8 



• Experimental for 2 -in Prism 

o « " /-/A 

I [^ 

- JoAC curves \f 

i u 



o. 07 -in. per day 

■ 0.035 -in. 2 • " 

f^oo -2ff00*/0~ tf 

Dashed curves if *o. 032 -in per day 
\k * o.o zo -in. 2 u 




Z 4 6 8 10 12 

Square Root of Dans Drying _ Vt~ 
Thickness fn inches b 

Fig. 16 Comparison of observed and calculated course in shortening and warping of 
prisms of mix A and cement M 



The curves in the upper diagram and the solid d the i 

diagram were constructed from the theoretical equations using the 
same values of the constants S », /, and k as were used in construe 
the curves of Fig. 15. Therefore, the three curves in the upper diau 
of Kg. 16 represent the same equations as three of the curves in Fig. 15; 
they differ only in the abscissas. The solid curves in the lo jram 

deviate considerably from the plotted point-. This indicate 
greater disagreement with theory than is shown in Kg. 15. II 
the dashed curves obtained from the theoretical equation by using the 
Bame valtie of Sea but with / reduced by 54 per cent and k reduced by 
43 per cent are in very good agreement with the experimental values. 



366 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

The computed shrinkage stresses will be about the same whether the 
first or the reduced values of/ and k are used. 

The values of the constants £», /, and fc, used in constructing the 
dashed curves, were obtained from three measurements as follows: (1) 
maximum warping of the 1-in. specimen, (2) time at which this maximum 
warping occurred, and (3) final shortening of a companion specimen. 
The agreement throughout the course of drying between the experi- 
mental values for warping and those given by the theoretical equations 
when these constants are used is excellent for the 1-in. specimen and very 
good for the 2-in. specimen. 

The above shows that if data on warping and data on shortening 
are analyzed separately, either group of data will appeal- to be in accord 
with the theory if the thicknesses of the specimens do not differ too much, 
but the values of/ and k obtained from the two groups of data will be 
different. The fact that the factor k is an empirical rather than a fun- 
damental property of the material is believed to be the chief reason 
why both groups of di * nted satisfactorily by on< 

nits. The empirical nature off is considered to be of only 
e in this study because it has much less effect than 
■time arid warping-vs.-time relations. 

Effect of differences in k on warping and shortening 

diffusivity on the theoretical value- of 

ping (Equ 2 bortening (Equation 2:>> : are shown in Fig. 

17 v ' plotted against the parameter V// jTfor 

■ ■■■ fc. As shown, differences in k have 

ply warping; each curve follows the same 

maximum point. The lower k t the greater 

m maximum value of warping, of 

ticipated because of the effect of k through tb 

As shown by the curves for 

I upon shortening is entirely different 

- The rate of shortening is materially reduced 

n the maximum shortening is unaffecfc 

i rent effects of changes in k on 
irping has been useful in explaining differences in 
made with cements of different composition. 
I results from concretes made with two diff- 
- - Mix C was used. A comparison of 
7 e conclusion that the coefficient of di- 

nt No. 5-1500-1.9 IS loner than for con- 






SHRINKAGE STRESSES IN CONCRETE 367 

cretes made with cement No. 1-1500-1.9*. Concretes made with cement 
from clinker No. 5 shortened at a lower rate but according to data not 
plotted eventually shortened more than concretes from clinker No. 1. 

Before this explanation was found, it seemed surprising that of two 
groups of specimens drying from one side only, subjected to the same 
exposure, one group would warp more and shorten less than the other 
group. In order that one specimen warp less than a second when the 
two specimens have the same average shrinkage, the distribution of 
shrinkage in the first specimen would have to be more nearly uniform. 
For the same surface conditions, a large value of k through the para- 
meter/6 k, tends to make shrinkage more nearly uniform and therefore 
is accompanied by less warping. An increase in uniformity of shrink- 
age also reduces the shrinkage stresses in an unrestrained specimen 
and therefore reduces the tendency for spontaneous cracking. (Fig. 14 
Part 1 — shows how the theoretical maximum stresses depend on the 
parameter fb/k (= B) .) 

Effect of alkali content on k and its possible effects on cracking 

It had been observed from various laboratory tests designed to measure 
cracking tendencies that concretes made with cements from clinker 
No. 5 tended to crack more than those made with cement from clinker 
No. 1, even though measurements often showed less volume change at 
the end of a given period of drying for the concretes of clinker No. 5. 
This greater cracking tendency of cement from clinker No. 5 was at- 
tributed to its higher alkali content, since this appeared to be the only 
important difference in their chemical compositions. Attempts to 
evaluate k for concretes made with cements from these two clinkers 
showed that for the same mix proportions the value of k for concrete 
made with the cement of higher-alkali content was only one-half that 
made with the cement of lower-alkali content. These observations sug- 
gested the possibility that: alkali reduced k, a reduced k resulted in 
higher shrinkage stresses, and higher stresses resulted in more cracking. 

To investigate this effect of differences in alkali content more fully, 
several tests were made using cement No. 1-1665-2.48. The procedure 
was to add 0.91 per cent Na 2 by weight of cement in the form of NaOH 
to the mixing water of one of two companion mixes. The results of one 
using mix B are shown in Fig. 19 where shortening and weight losses 
of prisms are plotted against period of drying. 

The dimensions of the prisms were 2^x23^x1134 in. They dried from 
all surfaces except the ends. By using for the specimens containing added 
alkali a time-scale equal to one-third the scale used for the regular 
specimens the corresponding curves for both sets of specimens approxi- 

*As explained in the Appendix, the first number is the clinker number, the second is the specific surface 
| Wagnea mel hod), and the third is the percentage of S0 3 . 



368 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 



Fig. 18— Warp- 
ing and shorten- 
ing of prisms that 

differ primarily in 
the alkali con- 
tent of th« 
ment used 

Prisms 3-in. thick in 
the direction of mois- 
ture travel. Deflec- 
tion measured over a 
32-in span. Mix C. 
Cured 7 days. 



Fig. 17 — Theoreti- 
cal effect of k on 
the course of con- 
traction and warp- 
ing of prisms dry- 
ing from one side 
only 




4 6 8 10 12 I4 


le 


2 4 6 8 


10 


Square Root of Number of Days Drying 






20 30 40 50 60 

Period of Drying -days (without added alkali) 
6C 90 120 ISO I6C ?io 

Penod of Drymg-days (alkali added) 



Fig. 19— Shrink- 
age and weight- 
loss for speci- 
mens with and 
without added 
alkali 



SHRINKAGE STRESSES IN CONCRETE 369 

mately coincided, indicating that the main effect of the added alkali 
was to reduce the diffusivities for both shrinkage and moisture flow to 
one-third the value without added alkali. 

The effect of added alkali in reducing the diffusivity of shrinkage 
for cement of clinker No. 1 is in accord with data reported by Haeger- 
mann. 21 Haegermann was primarily interested in the effects on shrink- 
age of additions of various sulfates to cements of different C 3 A contents. 
The sulfates tried were ferrous, calcium, magnesium, sodium, and po- 
tassium. The amounts added were such as to increase the S0 3 , content 
1 per cent, based on the cement. Five cements ranging from 15 per cent 
computed C Z A content to zero per cent C 3 A were investigated. 

The data were presented by Haegermann in the form of curves. For 
each cement, the curves representing the sodium and potassium sulfate 
additions are of noticeably different shape from the other curves for the 
same cement, the difference in shape being such as would result from a 
lower diffusivity. Since Haegermann did not give data on loss in weight 
during drying, it can only be inferred from the data on shrinkage that 
the sodium and potassium sulfates also reduced the diffusivity of moisture 
flow. 

From theoretical consideration, it appears that any highly soluble 
material should reduce the relative rate of drying; i.e., should increase 
the time required to lose a given percentage of the total amount of 
moisture to be lost.* However, since many other factors affect the 
rate of shrinkage, and alkalies have many other effects which may in- 
directly affect shrinkage, one should expect many real and apparent 
contradictions to the above indication that an increase in alkali content 
will retard shrinkage. 

The effect of the added alkali on cracking was investigated by means 
of the "wedge test"t and by the " restrained-shrinkage test" (subse- 
quently described). The result was that specimens of higher alkali 
content showed a much greater tendency to crack, as measured by these 
tests. 

Other tests made in this laboratory show that for cements containing 
an appreciable percentage of tricalcium aluminate, an increase in alkali 
content will increase final shrinkage of laboratory specimens unless the 
increase in alkali is accompanied by an increase in gypsum. The greater 

♦This reasoning is based on the supposition that at least part of the flow of water in concrete is by means 
of the following cycle: evaporation at an air-water interface, vapor diffusion across air space, capillary flow 
?n feJdfSEd Space and again evaporation at air-water interface. .Since the diffusivity of the soluble ma- 
terial within the liquid is finite rather than infinite, at any air-water interface at which water is evaporating 
ISL of soluble material will be higher than that for equilibrium with the adjacent liquid and 
therebv tend to restrict evaporation at this interface, and at any air-water interface at which water is con- 
densing the conrei.tration of soluble material will be lower than that for equilibrium with the adjacent 
Houid and therebv ten<l to restrict condensation at this interface. Therefore any highly soluble material 
should retard the'drving bv reducing the diffu-ivity of moisture flow. .„.,-.,. „ , 

t A speninei! L cast 5f1 he form of a wedge and. after curing, is permitted to dry from the two non-parallel 
surfaces. 



370 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

tendency to crack of the specimens with higher alkali content might 
have been due, at least in part, to a decrease in diffusivity and an in- 
crease in final shrinkage. 

On the other hand, the possible benefits from alkali should not be 
overlooked. The lowered rate of moisture loss will permit the interior of 
concrete to retain sufficient moisture for additional hydration for a longer 
time after drying of the surface begins. Prevention of complete drying 
of the interior during the usual drying season should be especially ad- 
vantageous in preventing cracking when restraints against shortening 
are present. Tests in this laboratory have also shown that concretes 
of higher alkali content have greater capacity for plastic flow, which is a 
favorable property. 

Decrease in k as drying proceeds 

As drying proceeds, the value of the coefficient of shrinkage diffusivity 
t apparently decreases. This decrease no doubt results from a pro- 
gressive decrease in the apparent diffusivity of moisture, diffusivity of 
moisture probably being a function of the moisture content. If diffusi- 
vity of moisture is a function of moisture content, then the shrinkage 
diffusivity can be considered to be a function of the shrinkage S and the 
differential equation becomes non-linear. Adding particular solutions, 
as was done in Part 1, is then not permissible. 

However, if in place of considering k to be a function of the dependent 
variable S it is considered to be a function of the independent variable 
t and of the dimensions of the body, then the differential equation re- 
mains linear. Furthermore, if the factor/is considered to varv with time 
in a hke manner so that the ratio f/k remains constant (see, for example 
lation 2a), then all of the equations for displacements, stresses and 
' - developed in Part 1 still apply if the symbol t appearing in then, 
d by a function of t and the dimensions. The changes suggested 
•■'iH.ve amount to a , ontinual change in the time-scale so that the time 
squired for given conditions to develop becomes progressively longer 
- ,n,,dl,vl "« the theory in this way better agreement with experimental 
results can he obtained. 

Fig. 20 is an example of applying the foregoing analysis. The plotted 
points are from experimental data on the average warping of four 3-in 
specimens ol concrete of mix B with cement 1-2280-]. 94. When an 
attempt „ras made to select constant values of/, Ife, and .<J. to be used 
"' "■ theoretical equation that would give curves in agreement w,.h all 
-•> the experimental values, no( all 1he (1:((a ( , )ljl(i 1)( , bK)ughl ^ 
men wnh the theoretical equation. But by taking the following 
lor the factors, a Letter fit was obtained. 



SHRINKAGE STRESSES IN CONCRETE 



371 



0.06 


































0-02 
n nr\ 



































a 



2 3 4 S 6 

Square Root of Dags Drying VT 
Thickness in inches b 



Fig. 20 Com- 
parison of theo- 
retical and ex- 
perimental warp- 
ing 

Points are from the 
average warping of 
four 3 - i n beams. 
Span 32-inches. Mix 
B. Cement 1-2280 
1.94. Cured 7 days. 



ft = 0.10-J^j^inVday 



■fh 

f = 1.67 ft in dav, i.e., - = 5 
k 

Soo = 765 x 10' 6 

When these values are introduced into the differential equations and a 

solution made, the symbol T in the final equations for warping, etc., is 

replaced by 

±k 



k [ 



2 + t 



1 



b 2 L^ 2 

where k is the initial value of k or 0.10 sq. in. per day. For convenience in 

making computations preliminary to plotting of the theoretical curve, 

b 2 T b A T 2 
t was expressed in terms of T, or t = 1 . The tabular values 

used for constructing the curve are given below : 



From Table 4 


Computed Values 




2bv max 




<t 




T 


3M» 


i 


b 


V max b 


0.01 


0.0152 


1.00 


0.333 


0.0178 


0.03 


0.0326 


3.61 


0.635 


0.0384 


0.10 


0.0538 


li). 1 


1.46 


0.0634 


0.15 


0.0557 


36.4 


2.01 


0.0655 


0.20 


0.0539 


58.5 


2.55 


0.0635 


0.30 


0.0470 


118 


3.62 


0.0553 


0.50 


0.0336 


298 


5.75 


0.03 


0.75 


0.021U 


639 


8.41 


0.0258 



The better agreement that can be obtained by the modified theory 

probably would not compensate for the extra work in all cases. Since the 



372 



JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 



Fig 22 Shrink- 
oge of slabs of 

n e s s • s drying 
from two oppo- 

des 







Fig. 21 Com- 
parison of theo- 
retical and ex- 
perimental warp- 
ing of beam ab- 
sorbing moisture 
from one side 

Poinli MPMHl th« 
warping ol a 1 -in 
b»om, having all but 
on* of i»t *urfac*s 
»»ol»d, during »ub- 
mtnion in water 
Cured 7 da yi and 
rh»n dn#d at bO 
R H for 10 month • 
b*for« Ivilmg Mi* 
6 C»m»nl 1-9980 
1 94 Span3Jinchei 




' 



.l.|.- 

the valm 

Worp,ng ol prisms during absorption of moisture from one s.de 



SHRINKAGE STRESSES IN CONCRETE 373 

zero warp where it remained for the rest of the test, a period of one 
month. The experimental value for the warp after 6 hours' exposure is 
considered to be in error. Other tests on 1-in. prisms did not show the 
indicated large decrease in warp between the fourth and sixth hours of 
exposure. 

Although the amount of experimental data on swelling is yet small, 
the indications are that the application of the theory as developed in 
Part 1 is limited, first, because at the beginning of wetting the moisture 
already present will ordinarily not be uniformly distributed; second, 
because of having remained wet longer, the cement in the interior re- 
gions will have hydrated more than that closer to the drying surface; 
and third, as the concrete becomes wet again, hydration again starts. 
Agreement with diffusion theory is not expected while hydration is 
occurring at an appreciable rate, especially if the formation of hydration 
products causes expansion. 

Effect of thickness on rate and amount of shrinkage of walls or slabs 

In an investigation of the effect of wall or slab thickness on the rate 
and amount of shrinkage the results shown in Fig. 22 were obtained. 
The specimens from which the data were taken were made of mix B. 
Cements of two different compositions and a fine and a coarse grind of 
each are represented. The specimens were cured seven days under 
water. The specimens were 34 inches long and of either 2x6- or 2xl2-in. 
cross section. By sealing all but two surfaces the prisms were made to 
represent slabs or walls of 2-, 6-, and 12-in. thicknesses drying from two 
opposite sides. For example, the specimens that represented a wall 12 in. 
thick were 2xl2x34-in. and dried from only the 2x34-in. surfaces, there 
being 12 inches between these surfaces. 

Sets of gage-points were cast in these pseudo slabs so that the short- 
ening over three or four 30-in, parallel gage lines could be measured on 
each specimen. Details are shown in the Appendix. Each curve was 
obtained by averaging the results from four specimens of a kind. 

As shown in Fig. 22, the results from these concrete specimens are in 
general similar to those obtained on the cement-silica mix discussed 
previously (see upper diagram, Fig. 16.). The curves have the charac- 
teristic S-shape found for similar plotting of data from smaller specimens. 
The thicker the slab the greater its fb/k ( = B) and, according to theory 
as shown by Fig. 8 (of Part I), jthe greater the shortening should be for a 
given value of the abscissa, ^t/b. The experimental data are partly in 
agreement and partly in disagreement with the theory in this regard. 
In the middle, straight-line portions of the curves, the curves are in the 
correct positions relative to each other, but in ever y case the relative 
positions become reversed at larger values of Vf/6. Also, the relative 



374 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

presenting the two < rinds are rev< 

- portion of the curves. This latter deviation 
on the basis of a non-linear stress-flow relation- 

- the negative, the specimen will 

>uld if plastic Bon did not take place. Of 

and negative Bows are equal (algebraic 

the specimen is not changed by plastic Bon 

in an unrestrained Bpecimen if the 

However, if the stress-flo* relation- 

bhe range of stresses developed, the 

than the first power of th< 

o( tin specimen is reduced by pi 
pronounced foi the thick< 

I Jurve H of Fig i i I 

the 
'!..• basu of a aon-lin< 
ol the Brat p 
rind* wei 
-„,i enti 

rig the I 

othei teste with 

Plorf.c How 









• 






SHRINKAGE STRESSES IN CONCRETE 375 

In the usual measurements of plastic flow the quantity measured is 
the inelastic deformation of a body that results from applied loads. 
From these measurements computations are made of the inelastic defor- 
mations of the individual elements, i.e., average unit deformation if 
the load is axial or unit deformation of the outer fiber if the load pro- 
duces flexure. If stresses from other sources are not present, the com- 
puted values may be representative of the actual plastic flow. But if 
stresses from other sources are present, the computed and actual values 
may differ appreciably. Therefore, if in addition to load stresses a 
specimen is under stress as a result of non-uniform temperature or non- 
uniform shrinkage, it should be made clear whether the term plastic 
flow refers to the resultant plastic flows of elements or to only computed 
plastic Bows produced by loads. Since the effects of load and the effects 
of < Irving are not simply additive, there is no clear basis for deciding 
how much of the total deformation is due to the stresses arising direct!} 
from the load. In agreement with previous writers, the deformations 
produced by loads will be taken as the difference between the deforma- 
tions of loaded specimens and the deformations of identical specimens 
under the same drying conditions but not under load. Only the defor- 
mations produced by loads will be computed and n I ><\ i 
but in the interpretation of results consideration will be given to what the 
actual inelastic deformations are believed to 

As shown by the formulas for plastic flow used in this paper, the t 
deformation produced by load is divided into tv ~, elastic and 

inelastic. The elastic part is considered to be that which would be 
recovered immediately if the load were rein- ed it is determined 
the computed load-stresses and the "dynamic" modulus of elastii 
The remaining part is considered to be the plastic How produced by the 
load. 

Some investigators make a slightly different division in that the 
elastic deformation is considered to be that which was product 
diately upon application of the load rather than that which would 
recovered immediately upon removal of the load. The two \ 
equal if the modulus of elasticity does not change during the test. Some 
writers prefer to divide the total deformation produced by load into I 
parts: ( 1) that recovered immediately upon removal of load, (2) that not 
immediately but eventually recovered, (3) the permanent deformation. 
McIIcnrv -' restricts the use of the term plastic flow to the third part. 
This division into three parts has merit, especially for those 
which the second part is an appreciable percentage of the total. For the 
data given in this paper no separation of the second and third parts 
could be made, but the permanent deformation (3) is believed to be 
much greater than the temporary (2). 



376 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

If the specimen is not shrinking while it is under load, then a con- 
siderable part of the inelastic deformation is probably only temporary 
and apparently the result of viscous flow in the adsorbed water films. 
After removal of the load, the elastic constituents of the gel-structure 
tend to restore the original shape but are retarded by the viscosity of 
the adsorbed water films. However, if an elemental volume* of cement 
paste is shrinking while under stress, the conditions are different. The 
loss of moisture introduces relatively large interparticle forces which 
tend to change the relative positions of the colloidal particles within an 
element. Some adjacent particles are pulled closer together but others 
are moved further apart. During this time of movement the directions 
of relative motion of the particles may be appreciably affected by stn 
on the element. In this v ses on an element during the time it is 

shrinking may produce comparatively large permanent deformations. 

The foregoing is one explanation for the much larger amount of 
plastic flow that a load will produce on a drying specimen compared to 
what it would produce if either tin- specimen were prevented from drying 
or had pr< been dried. It is also an explanation of the relatively 

• near the drying surface to deform plasti- 
cally without cracking. I) the analysis is correct, then a definite stn — 
flow relationship cannot l»< given element of concrete since 

vould depend not only upon the magnitude 
duration i on an element but also upon the changes in mois- 

d while the element was under stress. 

formation was considered to be per- 
i in deformation with time was con- 
hiefiy by changes in distribution of shrinl. 

i ly small lag in time after the develop- 

!n "pla8tic Mow" rather than "creep" 

;>.i! t of t he deformation. 

hat way the pa-tic flows of the individual 

deformations ol the body as a whole. 

dal load the plastic flow in ten-ion or 

. the load is the difference in the a um 

[nation ol each dement and what the algebraic 

body bad not been under load. But the pis 

ral load depends on the moment of the inela 
■ lement with neutral axu " Both 

aon side and plastic compression on the 
.1 load contribute to the measured 






' 






SHRINKAGE STRESSES IN CONCRETE 377 

Summary of remarks on plastic flow. The actual plastic deformation 
of elemental volumes of a specimen may be much different from that 
computed on the basis of laboratory experiments if shrinkage-stresses 
are present, but in this paper the plotted curves represent such computed 
values. Computed values are based upon the difference in the deforma- 
tions of loaded and not loaded specimens. The term "plastic flow" is 
used in this paper to refer to either actual or computed plastic deforma- 
tion. Plastic deformation is arbitrarily defined as that part of the total 
deformation produced by stress (either by actual stress or by load-stress 
as indicated by the text) that would not be immediately recovered upon 
removal of the stress. 

Effect of thickness on stresses and plastic Flow when the slab is partially restrained against 
shortening 

Companion specimens of the same size and sealed in the same manner 
as those represented in Fig. 22 were partially restrained against shrink- 
age by specially designed steel bars, somewhat as were those described 
by Carlson. 23 The main features of the steel bars are shown in Fig. 23. 
(The concrete specimen illustrated in Fig. 23, however, is from another 
test in which the concrete was allowed to dry from all sides and only one 
bar was used per specimen). Each specimen of 2x6-in. cross section con- 
tained two %-m. diameter bars, and each specimen of 2xl2-in. cross 
section contained four 5^-in. diameter bars. The arrangement of bars 
is shown in the Appendix, and in Fig. 24. 

A rubber tube covered the central 20 inches of each bar so as to prevent 
bond over a 20-in. gage length, thereby insuring the same axial force 
in the bar over all sections of the gage length. That part of each steel 
bar not covered with rubber was threaded and thus the bars were anchored 
to the concrete for a distance of 7 in. on each side of the gage length. Be- 
cause of this anchorage the shortening of the steel bar over the gage 
length is equal to the shortening of the concrete over the same gage 
length. Moreover, as is obvious from considerations of equilibrium, 
the force in the concrete in this gage length is equal and opposite to the 
force in the steel in the same gage length. Therefore, the average unit 
stress in the concrete can be computed from the change in length, modu- 
lus of elasticity, and percentage of steel. The formula is 

A S E S M 



(Jr = 



A c I 



where <x c is average stress in the concrete 

A 9 is cross-sectional area of the steel 
A c is cross-sectional area of the concrete 
E 8 is Young's modulus for the steel 



378 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 



n ( 


dUb 


-vHHIHHK 








• [ 


fS 




c 
01 

E 

u 

V 

a 

(/) 

r 
c 

N 




CO 

5 1 


i 


W 


\ 


\ v 1 


^r 


-0 9 — 9-*T 




_ * 8 ° S 






— Mr i 




SP«Z P/,;# 



■ V/ £ 2Qn± J*qqn# yjoouj? 




i = 



i\ 



o 



*■_ «t > 






I 



<3 

■ it 

c . .'C 

■z iz 9- 



SHRINKAGE STRESSES IN CONCRETE 379 

A/ is net change in length over gage length after corrections 
have been made for any change in temperature.* id is nega- 
tive if the specimen has shortened. 
/ is gage length. 
Not only the average stress caused by the restraint but also plastic 
flow caused by this stress can be computed if the modulus of elasticity 
of the concrete is known and the assumption is made that the shrinkage 
tendencies of the restrained specimens are the same as those for com- 
panion unrestrained specimens of the same size. The formula is 

° = S " -B. + 1 

where c is unit plastic flow caused by restraint, 

S av is the unit shortening of the free-shrinkage specimens, and 
E c is Young's modulus for the concrete. 

Performance of partially restrained specimens. As explained in the 
Appendix all specimens were cured under water. The specimens tended 
to expand during this storage and consequently the concrete in those 
partially restrained with restraining bars was compressed. Therefore, 
for a short time after drying began, the direction of the plastic flow 
produced by the restraining bars was in a negative direction. Shortly 
after drying began, the stresses in the restraining bars changed from 
tensile to compressive, and the average stress in the concrete changed 
from compressive to tensile. 

Under the conditions of this test the average stress reaches a maximum 
and then slowly decreases if failure by spontaneous cracking does not 
occur. A specimen's average stress and its shortening necessarily reach 
their maximums simultaneously if the temperature remains constant. 
Therefore, the time of maximum average stress is the time when the 
rate of average shrinkage equals the rate of plastic deformation. During 
the decrease of average stress, the rate of plastic deformation exceeds 
the rate of shrinking. 

Ordinarily in this test the specimens are not permitted to reach a final 
equilibrium state in regard to shrinkage, shrinkage-stress, and plastic 
flow. But just after the maximum restraining force has been developed 
additional tensile load sufficient to cause failure of the specimen is 
applied. This load is applied to the protruding threaded ends of the 
restraining bars by a machine designed for the purpose. While the load 
is being applied, measurements are taken so that the added stress in 
the concrete can be determined. Further details are given in Fig. 33 
of the Appendix. 

♦All tests were conducted in a room maintained at 76 =±= 1°F and a relative humidity of 50 =•= 2%, 
except for occasional deviations from these limits. # 



380 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

500, , m 

Fig. 25— Plastic 
flow in the par- 
Hall/ restrained 
specimens of Fig. 
24 



Cement 1-2260 -/. 94 




Factor of safety. The purpose of the testing just described is to 
learn how close the specimen comes to cracking spontaneously The 
ratio of the computed stress at failure to the maximum average shrinkage- 
stress is called a factor of safety. Specimens that crack spontaneously 
are reported as having a factor of safety Less than unitv. Results showing 
computed average shrinkage-stresses and plastic deformation are shown 
in Fig 24 and 25, respectively. The record of the number of specimens 
of each cement that cracked spontaneously and of the average factors of 
safety (f.s.) of those that did not crack is also shown in Fig. 24. 

Effect of thickness on plastic flow. Attention is called to the similarity 

ol the three sets of curves in Fig. 22, 24, and 25. The similarity is not 

to be interpreted warily indicating that the plastic deformation 

is proportional to its stress. One is tempted to make this 

1 " t " , , , 1 "V e » « it were true, then the plastic flow of a specimen 

would depend only on average si ress and not on the distribution of stress 
In general, the diagrams show that the 12-in. specimens developed con- 
siderably more plastic flow for the same amount of average stress than 
«<* -- or the 6-in. specimens. 

T1 >i possible reasons fortius: first, because of the lower 

temng, the larger specimens will have been under a given 

- longer than the smaller specimens and therefore would be 

expected to have more plastic flow for the same stress. Since the time 

^quired for the same amount of shortening is approximately pro, 

" i:i <" "'<• 7— - the thickness, the 12-in. specimens will in genial 

have been under a given range of stress about four times as long as the 

lH ": ~»™- Second, sin,,, the thicker specimens will have higher 

maxnnum etresses, the additional plastic flow could be accounted for 

by ■ non-hnear stress-flow relationship whether or not this reTtionsh* 






SHRINKAGE STRESSES IN CONCRETE 



381 



for each element was modified while the element was losing moisture 
rapidly. Third, the assumption that shrinkage tendencies of the re- 
strained-shrinkage and free-shrinkage specimens are equal is not entirely 
correct and consequently their computed plastic flow are in error. A 
difference in the shrinkage tendencies of the 12-in. free and restrained 
specimens might result since the arrangement of the four bars was such as 
partially to obstruct the flow of moisture. 

Probably all factors listed above contributed to the results. Of the 
factors causing the computed plastic flow to be greater in the 12-in. 
specimens, the author is of the opinion that the non-linear stress-flow 
relation contributed much more than the difference in duration of given 
stresses.* That the maximum stresses in the larger specimens are 
higher is shown by the fact that all the 12-in. restrained specimens of 
three of the four cements cracked (see Fig. 24) whereas only a few of the 
6-in. specimens and none of the 2-in. specimens cracked. 

According to most of these arguments the 2-in. specimens should 
have less plastic flow than the 6-in. specimens, whereas in general they 
have slightly more for the same shortening and for the same average 
stress. A complete explanation for this is not at hand, but the lesser 
extent of hydration of the cement in the 2-in. specimens because of their 
more rapid drying may be a factor. Also, the exposed surfaces of the 
2-in. specimens were the top and bottom surfaces as cast, whereas the 
drying surfaces for all the other specimens were the sides as cast. Bleed- 
ing and settlement of the plastic mix before initial hardening 
plete always makes the concrete near the top and that near the bottom 
as cast different from that at the sides. It must also be remembered 
that the computed plastic deformation may be more or less than the real 
plastic deformation of the material. 

Use of beams drying from only one side for determining probable stresses in slabs or walls 
drying from two opposite sides 

To obtain information on plastic flow and on the magnitude and 
distribution of stresses in unrestrained walls or slabs drying from two 
opposite sides another set of specimens, also companion to those repre- 
sented in Fig. 22, were made. These specimens differed from those of 
Fig. 22 in that they were permitted to dry from only one side in- 
of two opposite sides and in that the thicknesses of corresponding speci- 
mens were just half those of Fig. 22. Since they were half as thick and 
dried from only one side instead of two sides (see Appendix), any one of 

se specimens was considered to have the same conditions of di 
and consequently the same distribution of shrinkage tendency as either 
half of a corresponding specimen represented in Fig. 22. 

♦Most contemporary writers on the inelastic properties of concrete apparently would take the opposite 
view. ThS difference "in viewpoint » explained and an argument for the author's view is given in Ref. 20. 



382 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

If the distribution of shrinkage tendency is the same and if the correct 
external forces are applied so as to make all the deformations the same 
as those of either half of the corresponding specimen, then the distribu- 
tion of stresses will also be the same. 

Eight specimens of a kind were made, four of which were allowed 
to warp freely and four were restrained against warping. It was not 
feasible to distribute the external restraining forces on the specimens 
to be restrained against warping in exactly the manner that the mutual 
forces between the two halves of the corresponding specimens were 

ibuted. As shown in the Appendix, the method adopted ws 
support the specimen as a simple beam and to apply enough force at the 
quarter-points to prevent warping of the central half. 

Ifi discussed previously and as shown by Equation 19*, the amount 

of warping of a specimen free to warp is indicative of the non-uniformity 

linkage tendency. From similar considerations it follows that the 

sary to prevent warping is indicative of the non- 

uniformil ee resulting from the non-uniformity of shrinkage 

Furtli. the difference between the actual moment 

required ing and that computed from the amount of 

flow, gives an indication of the dis- 
tribution f ] one of the four cements are 

give the actual moment developed, 
livided I ection modulus, / c, for beams of three different 

what the Mc I would have been 

ment had not produced plastic flow. The upper 

measured values of the warping tendency 

and x ; panioo srx cimens. Foung's modulus was 

1 quency of vibration. 

^ ace the ordinatee in Pig. 26 are 

i at the stress in the outer fiber according 

Formula. According to the lower curves of 

or the 1 -in. specimens reached B maximum 

kefirs* day ol drying. Thecomputed 

3-in specimen was 190 lb. per sq. in. and 

mputed Btreas in the 6-in. specimens had 
and the indications are that, had the 
uued, the computed beds maximum 

• r about LOO days ol drying. om- 

reon the! 
The acta ■ the drj 

rapidly 



SHRINKAGE STRESSES IN CONCRETE 



383 



Fig. 26 — Com- 
parison of actual 
moment neces- 
sary to keep 
beams from 
warping with the 
conputed mo- 
ment necessary 
to strai g hten 
companion 
beams that are 
free to warp. 
(Plotted in terms 




will build up as rapidly in the thicker slabs as in the thinner. The 
actual stresses are probably better represented by the solid curves of 
Kg. 27. 

Stresses based on modified theory. The solid curves of Fig. 27 show- 
stresses based upon a modified theory. These curves were obtained by 
substituting appropriate values of the parameters y/b, kt/b 2 ,fb/k, E, and 
So into Equation 20 of Part 1. For the construction of these curves the 
theory as presented in Part 1 was modified in that, instead of using 
constant values for the factors k, f, Soo } and E, the following procedure 
was pursued: 

(1) The ultimate shrinkage S*> was set equal to 750, 700, and 600 
millionths, respectively, for the 1-, 3-, and 6-in. thick specimens. The 
selection of these separate values rather than one value for all specimens 
was governed by the apparent ultimate unit shortenings of the cor- 
responding free-shrinkage specimens (Fig. 22). 

(2) The ratio f/k was set equal to 2.5 in." 1 , i.e., fb/k was 2.5 for 1-in., 
7.5 for 3-in. and 15.0 for 6-in. specimens. When this value of f/k and 
the above values of &«, were used, the theoretical maximum values of 
warping as given by curve A of Fig. 14* were found to be in agreement 
with the experimental values of maximum warping for each of the three 
thicknesses of specimens. 

(3) A value of kt/b 2 was selected for each period of drying (1, 7, and 
28 days) and for each thickness of specimen, such that when substituted 
along with the above values of Sa> and f/k in Equation 23* for warping 
the result would be in agreement with the experimentally determined 
values for these periods and these thicknesses. 

(4) A value of E was selected for each period of drying and each thick- 
ness of specimen such that the theoretical moment given by Equation 
27 would be in agreement with the experimentally determined values. 

♦See Part 1. 



384 

Fig. 27 Theo- 
retical distribu- 
tion of shrinkage 
stresses (modified 
theory) in slabs 
drying from two 
opposite surfaces 
for various dry- 
ing periods and 
slab thicknesses 



JOURNAL OF THE AMERICAN CONCRETE INSTITUTE 
12 -in. Slab 6-in.5!ab 



W 



2000 

1500 

1000k 

500 



-500 

2000 

1500 

1000 

500 



-500 
2000 i 
1500 
1000 
500 




After Drying 1 Day 



•Theoretical stress based 
on dynamic £ 



^/Theoretical stress basea 

on reduced E 



> < [ ; I '_ 



\~ dynamic 



reduced 

J i 



\- dynamic 



After Drying 7 Days 




L— dynamic 

\ 
\ 
y* reduced 




A 



-500 L 



After Drying 28 Days 






dynamic 
reduced 



\T- 



">ahnic 



reduced 




2 3 4 S 6 12 3 

inches from Exposed Surface 



February 1946 

2-in.Slab 



— dynamic 

r 



r dynamic 



v 



^-dynamic 
\ (Educed 
\ 



I 



5 ' lues "• S , kt b\fb fc, and E together with appro- 

bstituted into Equation 20* and comp 
olid curves were plotted from bl 

sho* the stresses that arc obtained when the f< 
imputing is used except thai the dynamic 

E. The difference in th ua a 

by plastic How and thus is i .., 
!asti< deformation that baa occur* 

I thai for the most part the magnitude and 
es after the various | I drying and for the 

Besses ol specimens are about as given bj i 

ire probably in error in a Fheprin- 

"P* d error lies in the assumption thai the i 

aken Into account by using i reduced modulus of elasti 
- done when Equation 20 

ends oo the- past stress-hisi mJv M 

•See J 



SHRINKAGE STRESSES IN CONCRETE 385 

stress at the moment. Furthermore, as explained in the section on 
plastic flow, the rate of flow for an element will depend on the rate at 
which the element is tending to shrink and may not be proportional to 
the stress on the element. Therefore, near the drying surface, where 
the stress has been relatively high from the beginning of drying, the 
plastic flow will be greater and the stress will be less than that indicated. 
Slightly farther inward where the stress has only recently changed from 
compression to tension the resultant plastic deformation will be less 
and the stress more than that indicated. The dotted curve in the one 
diagram of Fig. 27 represents an attempt to show a better estimate of 
the actual stress. 

Reversal of stress by plastic flow. Fig. 26 indicates that eventually the 
stress in the outer fiber will become negative, i.e., compressive. In all 
restrained-warping tests that were continued until equilibrium of mois- 
ture content was nearly reached, the moment required to prevent warp- 
ing decreased to zero and would have then become negative if restraint 
against negative warping had been provided. This means that when a 
wall dries from two opposite sides or a prism dries from all four sides, 
eventually the outer shell will be in compression and the inner core will 
be in tension. 

Of interest in this connection is the fact that specimens of neat cement 
bars have been known to break spontaneously and audibly while 
resting in place in a storage rack. The explanation is that during the 
early part of drying large tensile stresses developed in the outer shell. 
As a result the outer shell was first permanently elongated and then 
caused to fail in tension, i.e., to crack. As drying proceeded inward, the 
inner core, which had not yet been stretched, tended to become shorter 
than the outer shell. The cracks closed, compression developed in the 
outer shell, and tension developed in the inner core. In some cases this 
tension was sufficient to cause failure of the core. A specimen would 
break spontaneously when failure of the inner core occurred at a section 
where the outer shell was already cracked. 

Investigation of properties of concrete by means of slabs or prisms drying from one side only 

As the foregoing has indicated, results from prisms drying from onl\ 
one side have been very valuable for ascertaining in what ways the 
theory of diffusion is applicable to shrinkage of concrete. They are 
also valuable for investigating certain properties of concrete, especially 
if used in connection with the diffusion theory. The chief advantage of 
drying a prism from only one side is that it tends to warp as well as 
shorten as it dries and thereby makes possible measurements not ob- 
tainable on prisms drying from all surfaces. 



386 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 



As explained previously, the results from prisms drying from only 
one side indicated a cause for concretes of higher alkali content to show a 
greater tendency to crack under some conditions. Additional results in 
regard to fineness of grinding and percentage of gypsum will now be 
reported for the information they give to illustrate how such prisms may 
be used to investigate properties of concrete. 

Effect of finer grinding on plastic flow. As explained previously, in 
addition to the one cement represented in Fig. 2G and 27, three other 
cements were tested at the same time and in the same manner. The re- 
sults for the other cements were similar in most respects to those shown 
in Fig. 26 and 27 for the one cement. There were some differences, how- 
ever. More plastic flow occurred in the specimens made with the two 
finer-ground cements. The differences in plastic flow with fineness of 
grinding in tin- ' similar to and in agreement with the difl'er- 

bh( se cements in the restrained-shrinkage test (Fig. 25). 

The effect of finer grinding on strength is similar to that produced by 
r curing and therefore since longer curing decreases plastic flow we 
migh that fim ag would also decrease plastic flow. How- 

bed in this way all Bhowed that finer grinding 
stic flow for both restrained-shrinkage and restrained- 

m thifl laboratory indicate that this 

:mcr grinding on plastic fio\i is indirect. A given quantity of 
i i a1 the time ol grinding is less effective in retarding the 
earl} -l reactions the finer the cement. 11 Lack of proper re- 

tardation tctions because of insufficient gypsum results 

shrinkage tendency and greater capacity for 
ccording to the indications, finer grinding, if 
percentage of gypsum, indirectly produces 
tendency to deform inelastically. 

oswered the question introduced in the discussion 
by the i • plastic flow did not also reverse the 

the finer-ground cement I ' ibly 
at tl trying the coarser-ground cements, becaua 

gth, flow more readily bul become lees pis 

at the later ages when the strengths are 

• talized. I -nice only slight differences in the 

I viate from the theory could aceount fol 

d since i from the theory in many differ 

possible answers to the question. 

ited in Rg. 24, according to which 
grind Ited in u -e- at the end of the curing 

produced by the the 



SHRINKAGE STRESSES IN CONCRETE 



387 



concrete to expand during curing, especially the tendency to expand 
after some resistance to plastic flow had developed. Any tendency 
for the interior of the free-shrinkage specimens to continue expansion 
after the surface begins to dry would reduce the rate of shortening at the 
beginning of the drying period. If, as seems quite probable, this tendency 
is greatest for the thicker specimens with the coarser-ground cements, 
then these specimens would shorten relatively less at the beginning of 
drying than would be indicated by theory. 

Effect of added gypsum. The effect of the gypsum in the cement on 
the properties of the hardened concrete was observed in an investiga- 
tion in which 21 cements were made from the five clinker compositions 
listed in the Appendix. By blending various grinds of these clinkers, 
cements of different finenesses and different gypsum contents were 
obtained from each clinker. Concretes (Mix C) made from these 21 
cements were tested in the manner indicated previously for prisms drying 
from only one side. However, in this investigation only the 3-in. size of 
specimen was used. 

Where the C^A content of the clinker was moderate or relatively high, 
an increase in SOz, content decreased shrinkage and warping and also 
decreased plastic flow. Where the C 3 A content was low, an increase in 
SOz had relatively little effect. According to other data obtained in this 
laboratory, a still further increase in SO z would have increased the 
shrinkage. 24 Representative results for a cement of high CzA content 
are shown in Fig. 28, 29, and 30. As shown by Fig. 28, the maximum 
warp of those specimens free to warp was reduced appreciably by in- 
crease in per cent of SOz- The reduction in shortening with increase in 
SOz agrees with that reported previously by other investigators 21 - 24 - 25 . 

Fig. 29 shows that the restraint developed by the specimens restrained 
against warping was in general less with the higher percentages of SOz- 
However, increasing the SOz from 1.5 to 2.4 per cent had only a very 
small effect on the amount of restraint developed in the restrained 
specimens compared to the effect on the warping of unrestrained speci- 
mens. The explanation is that although the increase in SOz reduced 
warping it also reduced the tendency to yield under stress. The net 
result is some reduction in stress but not as much as would be antici- 
pated from the results of the free-warping specimens. 

Fig. 30 shows the effect of SOz on the factor of safety against cracking 
as determined by this test of a cement of high CzA. 



SUMMARY AND CONCLUSIONS 



The theory that shrinkage of concrete follows the laws of diffusion 
similar to those followed by the flow of heat is tested by means of specially 



388 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 




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SHRINKAGE STRESSES IN CONCRETE 



389 



3.0 



2.6 



i ! r — i 1 — 


f r ■ t r r t 
7 days) j ^ 


- 


Cured 7 days \ 

Jp. Surface 1500^^ ■ 


~Zs^ X / 28 days 


- 


/ s^ 


/ ^^C/ 28 days 

^^ \ 2IOO 


_ 


_.. II,,, 


i.i,]; 


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Fig. 30 Effect 
of gypsum, peri- 
od of curing, and 
fineness of grind- 
ing on factor of 
safety against 
cracking 

Specimens same as 
those represented in 
Fig. 29. 



10 1.5 2.0 2.5 3-0 3.5 

5O3 Content of Cement, % by weight 



4.0 



designed experiments. According to the theory th developed in 

Part 1, the shrinking and development of stress in a given coni 
under given conditions of drying is consii be chai 

certain constants. These constants are diffusivity of shrinkage, suri 
factor, ultimate shrinkage, and Young's modulus of elasticity. Equa- 
tions were derived in Part 1 g I warping of prisma 
versus period of drying in terms of these constants and the dimensions 
of the prisms. 

In Part 2 it is shown that these con- m be seta 

shortening of a prism as computed by the theoretical equations 1- in 
good agreement with experimental values of shortening. Furthern 
it is >hou 11 that by using the same constants the shortening versus period 
of drying of other prisms differing in size and number of - 
to drying can be predicted with fair accuracy if the difference in si 
not too great. However, it is shown that the theory r modified 

to take into account inelastic deformation and to permit the supposed 
constants to vary with moisture id size of specimen if the 

theory is to be in agreement with all results on all t> mens 

of a given concrete. 

The theory is used to explain various things about concrete; in I 
paradoxically, it is used to explain some of th< which con< 

does not perform as predicted by tin- theory. The tendency of I 
specimens to crack more ami shrink Less than smaller specimens and the 
effect of alkali content of the cement in increasing the tendency to warp 
while reducing the rate of shrinkage are explained on th- the 



390 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

theory. It is shown that when a saturated specimen is dried at 50 per 
cent relative humidity the stress developed would be much greater 
than the strength of the concrete if it were not for the effects of plastic 
flow. It is further shown that when a specimen is restrained against 
deforming the restraining forces are much less than they would be if 
plastic flow did not occur. 

An example is given of the use of the theoretical equation in determin- 
ing the distribution of stresses at various times during the drying of a 
specimen. In this example, consideration is given to plastic flow and to 
the decrease in diffusivity of shrinkage as the specimen dries. 

The restrained-shrinkage test and the restrained-warping test are 
used to determine a factor of safety against cracking for concrete under 
conditions of drying and of restraint comparable to those under which 
the tests are made. These tests, together with tests on free-shrinkage 
and free-warping specimens, are used to measure plastic flow. 

The various tests described in Part 2 and in the Appendix to Part 2 
when used in conjunction with the theory given in Part 1 provide a 
means for studying some of the more fundamental properties of concrete 
and for predicting the performance of concrete under some conditions 
in the field. 

ACKNOWLEDGMENTS 

This paper developed during a study of the causes and the control 
of cracking of concrete. The author is. indebted to many present and 
past workers in this field. E. A. Ripperger, now Lt. (j-g.) U.S.N, with 
the Pacific fleet, was responsible for a large part of the experimental 
work reported herein. He designed (or adapted from earlier designs) 
most of the special equipment used and was engaged in certain phases of 
the study previous to the author's participation. All of the work was 
done under the supervision of F. R. McMillan, Director of Research, 
and T. C. Powers, in charge of Basic Research. 

The author is particularly indebted to Mr. Powers for assistance 
in preparing the manuscripts. His suggestions in regard to presentation 
of material and the wording of various paragraphs have been invaluable. 

The author also wishes to thank Miss Adele Scott for preparing the 
diagrams and Miss Virginia Atherton for proofreading the manuscript. 



SHRINKAGE STRESSES IN CONCRETE 



391 



APPENDIX TO PART 2 



Mix proportions 





Parts by Weight 




Mix A 


Mix B 


Mix C 




0.5 

0.5 
1.0 

0.6 


0.355 to 0.388 

0.048 
0.403 to 0.436 

1.0 

1.28 
1.82 


487 


Water added for absorption 


083 




570 




1 






Elgin sand 


2 43 




2 97 







Consistency 

Consistency of Mix B with different cements was maintained fairly constant at from 
5 to 6 in. of slump with a 12-in. cone by varying the amount of mixing water. The 
consistencies of Mixes A and C were allowed to vary with the different cements. Mix C 
usually gave a slump of from 2 to 4 inches, but with some cements the slump was as 
little as 1.5 inches and with others as much as 6 inches. 

Materials 

Cements: One cement designated M was a mixture of four brands of Type I cement, 
purchased in Chicago. Its specific surface by the Wagner method was 1665 sq. cm. 
per g. The other cements were prepared from five different commercial clinkers. From 
each of these clinkers cements of three different finenesses, coarse, medium, and fine, 
were prepared by grinding at the plant. In addition, two cements, one of low and one 
of high gypsum content, were prepared from each clinker by grinding in a small lab- 
oratory ball mill. The purpose in preparing these five different cements from each 
clinker was to make it possible to obtain any desired fineness and gypsum content by 
blending different grinds of the same clinker. In referring to these cements in the text 
the first number in the designation is the clinker number, the second is the Wagner 
specific surface, and the third is the per cent S0 3 content by weight. 

The chemical compositions of the five clinkers and of cement M are shown below. 



Oxides 



Cement Clinker No. 



Chemical Analysis, 
per cent by wt. (corr. for minor components) 



20.67 


23.05 


27.82 


5.48 


4.14 


1.93 


2.50 


4.35 


1.87 


65.00 


64.28 


65.38 


1.31 


1.36 


1.75 


0.19 


0.03 


0.17 


0.85 


1.05 


0.26 


2.71 


0.73 


0.23 


0.30 


0.05 


0.05 


0.40 


0.17 


0.22 



Cement 
M 



SiOi 

AhOa... 

Fe20 2 

Combined CaO 

MgO 

S0 3 

Loss on Ign. . . . 

Free CaO 

Na 2 

K 2 

Compounds 
C3S 

C2* 

I \ 

< YU , 



21.54 
6.52 
1.56 

64 . 32 
2,17 
0.41 
0.15 
0.98 
0.17 
0.16 



50.73 

23.49 

14.72 

\ 75 



22.56 
5.00 
2.48 

64 06 
3.35 
0.20 
0.37 
nil 
1.13 
0.44 



Computed Compound Composition 
per cent by wt. 



66.57 


52.37 


38.58 


51.61 


9.05 


26.58 


50.66 


25.75 


10.29 


3 61 


1.95 


9.06 


7.61 


13.24 


5.69 


7.55 



21.25 
5.98 
2.69 

62.56 
3.04 
1.75 
1.13 
0.79 
0.28 
0.63 



44.15 
27.62 
11.30 

^ I'.i 



392 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

Aggregate: Sand and gravel were from Elgin, Illinois. The gravel was screened 
to pass a %-in, sieve and be retained on a Xo. 4 sieve. 

The sand was graded as folL 



Sieve Per Cent 

Xo. Bine 

100 a 

15 

28 4:< 

14 

82 

4 100 

Pulverized Silica: The silica was from the same source as standard Ottawa sand, but 
ground to cement fineness. Its specific surface b)- the air-permeability method was 3200 
sq. cm. per g. 

Procedure 

With some exceptions the procedure in preparing and testing the specimens was as 
follows: 

Preparing the specimens: The materials were mixed in a power-driven open-tub 

mixer. The fractions of the various-sized aggregates and grinds of cement were weighed 

and placed in the tub. The mixing schedule was: mix ] _> minute dry, 2 minutes wet, 

mix 2 minutes. (Special tests showed that the grinds of cement were 

adequately blended in }/> minute of dry mixing in the proence of sand.) 

The freshly mixed concrete was placed in steel molds and consolidated by light 
vibration by placing molds on a platform-type vibrator. Covers were fastened on the 
molds but not made water-tight. Each mold was equipped with restraining bars, 
gage inserts, etc., the details of assembly depending upon the tests to be made. 

The molds and contents were stored under water at 74 F for one day. The molds 
were then stripped and the specimens returned to water at 76 =«= 1 F in a covered tank 
where they were left for one hour. The specimens were then removed one at a time, 
dried with a cloth, and all initial measurements of length, deflection, and weight were 
made. All dimension measurements were made within 30 seconds after removal from 
water, and weighings were made as soon thereafter as practical. The specimens were 
then returned to the 76 F curing tank. 

One day before the end of the curing period the specimens were removed for sealing 
of certain surfaces against the loss of moisture. The surfaces to be sealed were wiped 
with a cloth and then allowed to air-dry until the surface just changed color. During 
.me. scheduled measuren. usually made. After the color change and 

before any appreciable loss of moisture by evaporation, one coat of black, quick-drying 
brushing lacquer was applied. After the lacquer had dried a few minutes, the - 

■d with a cloth and one coat of hot paraffin was applied to the lacquered 
surface. While a second coat of paraffin was being applied, one thickness of t *- 
paper of appropriate shape wa- pressed into the still soft parafiin somewhat in the 
manner in which a paper hanger applies wallpaper. Next, a final heavy coat of paraffin 
ipplied. The layer of paper helped to eliminate pin holes. (When only the ends 
of prisms were Bealed, the paper and final coat were omitted.) 

r the specified surfa Jed. the unsealed surfaces and exposed steel parts 

were cleaned, vaseline was applied to the steel parts, and the specimen was returned to 
the curing tank for an additional day of curing. At the end of the curing period the 
specimens were transferred from the curing tank to a room maintained at 76 ± 1 F and 



SHRINKAGE STRESSES IN CONCRETE 



393 



50 ± 2 per cent relative humidity. At this time the vaseline was removed from the 
steel parts and measurements for the beginning of the drying period taken. 

Testing the specimens: The testing was considered to have begun in most of the 
tests with the beginning of the drying period. The specimens may be divided into 
classes, according to the tests made, as follows: 

Free shrinkage specimens were measured for length changes, weight losses and resonant 
frequency of vibration. Reference plugs were cast in the ends of the specimens for the 
length-change readings. They were hex-head cap screws arranged to give the desired 
gauge length. Ordinarily these were single plugs centrally located in the ends. But 
for those prisms that represented slabs drying from two opposite sides the arrange- 
ment was as shown in Fig. 31. 



o o o 



>i" \ii" \ti"\ f i" ] 

6" -| 



o 
o 
o 



sea/ed^\ 



6" 



h*H 



1 


Z" 




o 


2i» 




o 


.__ 










J- 




o 


H' 




o 


2* 





FH 



Fig. 31 — End views showing arrangement of gage plugs in specimens representing 2-in,, 6- 

in. and 12-in. slabs 



The gage plugs for these specimens were Y% in. bolts 4 in. long. 

Restrained shrinkage specimens were partially restrained against shrinkage by steel 
restraining bars. The arrangement of the bars in those specimens that represented 
slabs drying from two opposite sides was the same as that shown in Fig. 31 for gage plugs 
except that only two bars 3 inches apart were cast in the 2x6-in. specimens. Square 
specimens that are permitted to dry from all four sides and that are partially restrained 
by one centrally located bar are used in routine testing of resistance to cracking. Further 
details in regard to restraining bars and the measurements for change in length of the 
restraining bar are shown in Fig. 32 and 33 as well as in Fig. 23 body of the 1 1 

If the restrained-shrinkage specimen did not crack spontaneously before the maximum 
restraining force had been developed, additional increments of load were applied as 
shown in Fig. 33 until failure was produced. 

Free warping specimens were measured for deflection over a 32-in. span. Most of 
these specimens were also measured for length change, weight loss and resonant fre- 



394 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

quency. Some of the free warping specimens made in the early part of the work were 
not equipped for length-change measurements. Cross sections of free warping speci- 
mens are shown in Fig. 34. See Fig. 35. 

Restrained warping specimens companion to the free warping specimens were loaded, 
as shown by Fig. 36, so as to prevent warping between the loaded quarter-points. Read- 
ings of the load required were taken periodically and, after the maximum restraint had 
developed, the load necessary to produce failure was determined. 





Fig. 32a (left) Comparator 

Fig. 32b (above) ^Restraining Bar 



Measurements of shrinkage-stress are made whether cracking occurs or not. The 
comparator (in Fig. 32a) is used to measure the changes in length of the steel restrain- 
ing bar that result from the strains placed on it by the concrete. From these measure- 
ments Hid the known properties of steel, the average stress in the concrete is computed 

Important details of the restrained-shrinkage type of specimens are shown in Fig. 32b 
(See also Fig. 23 in the tea thai the o an grip the bar only in the end- 

region; contact intl 2 tion is prevented by a thick layer of rubber. 



SHRINKAGE STRESSES IN CONCRETE 



395 




Fig. 33a 



Fig. 33b 



This method of test has been used to measure shrinkage-stress in specimens as wide 
as 12 in. and containing as many as four restraining bars. 

Those specimens that do not fail under shrinkage-stress alone are given additional 
stress with the machines shown in Fig. 33a and 33b. The machine at the left (Fig. 33a) 
is used for most of the specimens; that at the right Fig. 33b is used if the capacity of the 
other one is exceeded and if the specimen contains more than one bar. The load is 
applied to the bar, and the extension of the bar at the time the concrete fails is determined 
by the strain-gage shown in the pictures. The net amount of load on the concrete at 
failure is computed from the magnitude of the load and the strain-gage reading. The 
factor of safety is the ratio of the load on the concrete at failure to the maximum load 
represent od by the restraint against shrinkagi . 

"Sonic" testing: Most of the "unrestrained" specimens are tested periodically for 
frequency of vibration with the apparatus shown in Fig. 35a. Young's modulus is 
calculated from the resonant frequencies. 



ill 



396 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 



sealed — -\ 



r 




r 

j" 



\-*--\ 




Fig. 34 — Cross section of 
free warping specimens. 




Fig. 35a Sonic testing 



Fig. 35b— Warping 
measurement 




SHRINKAGE STRESSES IN CONCRETE 



397 





Fig. 36a (above) 
Fig. 36b (left) 
Restrained warping 



398 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 

Warping, length-change, and moisture-content: On specimens like that shown in Fig. 
35a, the change in length is measured with the comparator shown in Fig. 32 and the 
warping is measured with the curvature-gage shown resting on a specimen in Fig. 35b. 
Results are correlated with concomitant changes in moisture-content. 

Warping due to swelling: Previously dried specimens are placed, uncoatod side down, 
in a trough of water as shown Fig. 35b or ( bey are exposed to saturated air. The result- 
ing warping due to absorption at one surface is more rapid than that due to shrinkage 
and can ^ive rise to larger stresses. 

The specimens shown Fig. 36 are supported only at the ends. They are coated on all 
hut the bottom side and therefore as t bey dry they tend to bow upward. This tendency 
i> opposed b) th< shackles ai tin quarter-points which are connected to a lever system 
below, one for each specimen. The levers are held by the fine-thread screw-adjustment 
seen best in Fig. 36b. The screws are turned dow award until the force is just sufficient 
to prevent warping as indicated by the curvature-gage shown in both pictures. This 
i be moved from specimen to specimen. 

The force on the lever is measured periodically by finding the weight (bucket of shot) 

that will just hold the specimen in the position maintained by the screw. The force 

required reaches a maximum and iln-n n cedes slowly to zero as drying continues. When 

the maximum is reached, i he sp< cimens are loaded to failure. The ratio of the maximum 

>:nn\ to pn ping to that required for failure Lb the factor of safety. 

the factor of safety against cracking of Blabs of twice 
of the spe< - i quail} from both sides.